AP® Statistics
2006 Free-Response Questions
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2006 AP® STATISTICS FREE-RESPONSE QUESTIONS
Formulas begin on page 3.
Questions begin on page 6.
Tables begin on page 12.
2
2006 AP® STATISTICS FREE-RESPONSE QUESTIONS
Formulas
(I)
Descriptive Statistics
∑ xi
n
x =
d
i
sx =
2
1
∑ xi − x
n−1
sp =
dn1 − 1is21 + dn2 − 1is22
dn1 − 1i + dn2 − 1i
y = b0 + b1 x
d
id
∑ xi − x yi − y
b1 =
d
∑ xi − x
i
i2
b0 = y − b1 x
r =
b1 = r
sb =
1
FG
H
x − x
1
∑ i
sx
n−1
IJ FG yi − y IJ
K H sy K
sy
sx
d
i2
d
i2
∑ yi − yi
n−2
∑ xi − x
3
2006 AP® STATISTICS FREE-RESPONSE QUESTIONS
(II)
Probability
P( A ∪ B ) = P( A) + P( B ) − P( A ∩ B )
P( A B ) =
P( A ∩ B )
P( B )
E ( X ) = µ x = ∑ xi pi
d
Var( X ) = s 2x = ∑ xi − µ x
i2 pi
If X has a binomial distribution with parameters n and p, then:
P( X = k ) =
FG nIJ p k (1 − p)n − k
H kK
µ x = np
sx =
µ
s
np(1 − p)
p
= p
p
=
p(1 − p)
n
If x is the mean of a random sample of size n from an infinite
population with mean µ and standard deviation s, then:
µx = µ
sx =
s
n
4
2006 AP® STATISTICS FREE-RESPONSE QUESTIONS
(III)
Inferential Statistics
Standardized test statistic:
statistic - parameter
standard deviation of statistic
Confidence interval: statistic ± (critical value) ∑ (standard deviation of statistic)
Single-Sample
Standard Deviation
of Statistic
Statistic
σ
Sample Mean
n
p(1 − p)
n
Sample Proportion
Two-Sample
Standard Deviation
of Statistic
Statistic
Difference of
sample means
σ 12
n1
+
σ 22
n2
Special case when σ 1 = σ 2
σ
Difference of
sample proportions
1
1
+
n1 n2
p1 (1 − p1 )
p (1 − p2 )
+ 2
n1
n2
Special case when p1 = p2
b g
p 1− p
Chi-square test statistic =
1
1
+
n1 n2
aobserved − expectedf
∑
expected
5
2
2006 AP® STATISTICS FREE-RESPONSE QUESTIONS
STATISTICS
SECTION II
Part A
Questions 1-5
Spend about 65 minutes on this part of the exam.
Percent of Section II grade— 75
Directions: Show all your work. Indicate clearly the methods you use, because you will be graded on the
correctness of your methods as well as on the accuracy and completeness of your results and explanations.
1. Two parents have each built a toy catapult for use in a game at an elementary school fair. To play the game,
students will attempt to launch Ping-Pong balls from the catapults so that the balls land within a 5-centimeter
band. A target line will be drawn through the middle of the band, as shown in the figure below. All points on the
target line are equidistant from the launching location.
If a ball lands within the shaded band, the student will win a prize.
The parents have constructed the two catapults according to slightly different plans. They want to test these
catapults before building additional ones. Under identical conditions, the parents launch 40 Ping-Pong balls from
each catapult and measure the distance that the ball travels before landing. Distances to the nearest centimeter
are graphed in the dotplots below.
(a) Comment on any similarities and any differences in the two distributions of distances traveled by balls
launched from catapult A and catapult B.
(b) If the parents want to maximize the probability of having the Ping-Pong balls land within the band, which
one of the two catapults, A or B, would be better to use than the other? Justify your choice.
(c) Using the catapult that you chose in part (b), how many centimeters from the target line should this catapult
be placed? Explain why you chose this distance.
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2006 AP® STATISTICS FREE-RESPONSE QUESTIONS
2. A manufacturer of dish detergent believes the height of soapsuds in the dishpan depends on the amount of
detergent used. A study of the suds’ heights for a new dish detergent was conducted. Seven pans of water were
prepared. All pans were of the same size and type and contained the same amount of water. The temperature of
the water was the same for each pan. An amount of dish detergent was assigned at random to each pan, and that
amount of detergent was added to the pan. Then the water in the dishpan was agitated for a set amount of time,
and the height of the resulting suds was measured.
A plot of the data and the computer output from fitting a least squares regression line to the data are shown
below.
(a) Write the equation of the fitted regression line. Define any variables used in this equation.
(b) Note that s = 1.99821 in the computer output. Interpret this value in the context of this study.
(c) Identify and interpret the standard error of the slope.
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2006 AP® STATISTICS FREE-RESPONSE QUESTIONS
3. The depth from the surface of Earth to a refracting layer beneath the surface can be estimated using methods
developed by seismologists. One method is based on the time required for vibrations to travel from a distant
explosion to a receiving point. The depth measurement (M) is the sum of the true depth (D) and the random
measurement error (E). That is, M = D + E. The measurement error (E) is assumed to be normally distributed
with mean 0 feet and standard deviation 1.5 feet.
(a) If the true depth at a certain point is 2 feet, what is the probability that the depth measurement will be
negative?
(b) Suppose three independent depth measurements are taken at the point where the true depth is 2 feet. What is
the probability that at least one of these measurements will be negative?
(c) What is the probability that the mean of the three independent depth measurements taken at the point where
the true depth is 2 feet will be negative?
4. Patients with heart-attack symptoms arrive at an emergency room either by ambulance or self-transportation
provided by themselves, family, or friends. When a patient arrives at the emergency room, the time of arrival is
recorded. The time when the patient’s diagnostic treatment begins is also recorded.
An administrator of a large hospital wanted to determine whether the mean wait time (time between arrival and
diagnostic treatment) for patients with heart-attack symptoms differs according to the mode of transportation.
A random sample of 150 patients with heart-attack symptoms who had reported to the emergency room was
selected. For each patient, the mode of transportation and wait time were recorded. Summary statistics for each
mode of transportation are shown in the table below.
Mode of
Transportation
Sample Size
Mean Wait Time
(in minutes)
Ambulance
Self
77
73
6.04
8.30
Standard Deviation
of Wait Times
(in minutes)
4.30
5.16
(a) Use a 99 percent confidence interval to estimate the difference between the mean wait times for ambulancetransported patients and self-transported patients at this emergency room.
(b) Based only on this confidence interval, do you think the difference in the mean wait times is statistically
significant? Justify your answer.
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2006 AP® STATISTICS FREE-RESPONSE QUESTIONS
5. A biologist is interested in studying the effect of growth-enhancing nutrients and different salinity (salt) levels in
water on the growth of shrimps. The biologist has ordered a large shipment of young tiger shrimps from a supply
house for use in the study. The experiment is to be conducted in a laboratory where 10 tiger shrimps are placed
randomly into each of 12 similar tanks in a controlled environment. The biologist is planning to use 3 different
growth-enhancing nutrients (A, B, and C) and two different salinity levels (low and high).
(a) List the treatments that the biologist plans to use in this experiment.
(b) Using the treatments listed in part (a), describe a completely randomized design that will allow the biologist
to compare the shrimps’ growth after 3 weeks.
(c) Give one statistical advantage to having only tiger shrimps in the experiment. Explain why this is an
advantage.
(d) Give one statistical disadvantage to having only tiger shrimps in the experiment. Explain why this is a
disadvantage.
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2006 AP® STATISTICS FREE-RESPONSE QUESTIONS
STATISTICS
SECTION II
Part B
Question 6
Spend about 25 minutes on this part of the exam.
Percent of Section II grade— 25
Directions: Show all your work. Indicate clearly the methods you use, because you will be graded on the
correctness of your methods as well as on the accuracy and completeness of your results and explanations.
6. A manufacturer of thermostats is concerned that the readings of its thermostats have become less reliable (more
variable). In the past, the variance has been 1.52 degrees Fahrenheit (F) squared. A random sample of 10
recently manufactured thermostats was selected and placed in a room that was maintained at 68°F. The readings
for those 10 thermostats are given in the table below.
Thermostat
Temperature (°F)
1
2
3
4
5
6
7
8
9
10
66.8 67.8 70.6 69.3 65.9 66.2 68.1 68.6 67.9 67.2
(a) State the null and alternative hypotheses that the manufacturer is interested in testing.
It can be shown that if the population of thermostat temperatures is normally distributed, the sampling
distribution of
(n - 1) s 2
s2
(b) Calculate the value of
follows a chi-square distribution with n - 1 degrees of freedom.
(n - 1) s 2
1.52
for these data.
(c) Assume that the population of thermostat temperatures follows a normal distribution. Use the test statistic
(n - 1) s 2
from part (b) and the chi-square distribution to test the hypotheses in part (a).
1.52
(d) For the test conducted in part (c), what is the smallest value of the test statistic that would have led to the
rejection of the null hypothesis at the 5 percent significance level?
Mark this value of the test statistic on the graph of the chi-square distribution below. Indicate the region that
contains all of the values that would have led to the rejection of the null hypothesis.
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2006 AP® STATISTICS FREE-RESPONSE QUESTIONS
(e) Using simulation, 1,000 samples, each of size 10, were randomly generated from 3 populations with
different variances. Each population was normally distributed with mean 68 and variance greater than 1.52.
(n - 1) s 2
for each population.
The histograms below show the simulated sampling distribution of
1.52
Mark the region identified in part (d) on each of the histograms below.
(f) Based on the regions that you marked in part (e), identify the simulated sampling distribution that
corresponds to the population with the largest variance. Then identify the simulated sampling distribution
that corresponds to the population with the smallest variance. Justify your choices.
STOP
END OF EXAM
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2006 AP® STATISTICS FREE-RESPONSE QUESTIONS
Probability
Table entry for z is the
probability lying below z.
Table A
z
Standard normal probabilities
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
– 3.4
– 3.3
– 3.2
– 3.1
– 3.0
– 2.9
– 2.8
– 2.7
– 2.6
– 2.5
– 2.4
– 2.3
– 2.2
– 2.1
– 2.0
– 1.9
– 1.8
– 1.7
– 1.6
– 1.5
– 1.4
– 1.3
– 1.2
– 1.1
– 1.0
– 0.9
– 0.8
– 0.7
– 0.6
– 0.5
– 0.4
– 0.3
– 0.2
– 0.1
– 0.0
.0003
.0005
.0007
.0010
.0013
.0019
.0026
.0035
.0047
.0062
.0082
.0107
.0139
.0179
.0228
.0287
.0359
.0446
.0548
.0668
.0808
.0968
.1151
.1357
.1587
.1841
.2119
.2420
.2743
.3085
.3446
.3821
.4207
.4602
.5000
.0003
.0005
.0007
.0009
.0013
.0018
.0025
.0034
.0045
.0060
.0080
.0104
.0136
.0174
.0222
.0281
.0351
.0436
.0537
.0655
.0793
.0951
.1131
.1335
.1562
.1814
.2090
.2389
.2709
.3050
.3409
.3783
.4168
.4562
.4960
.0003
.0005
.0006
.0009
.0013
.0018
.0024
.0033
.0044
.0059
.0078
.0102
.0132
.0170
.0217
.0274
.0344
.0427
.0526
.0643
.0778
.0934
.1112
.1314
.1539
.1788
.2061
.2358
.2676
.3015
.3372
.3745
.4129
.4522
.4920
.0003
.0004
.0006
.0009
.0012
.0017
.0023
.0032
.0043
.0057
.0075
.0099
.0129
.0166
.0212
.0268
.0336
.0418
.0516
.0630
.0764
.0918
.1093
.1292
.1515
.1762
.2033
.2327
.2643
.2981
.3336
.3707
.4090
.4483
.4880
.0003
.0004
.0006
.0008
.0012
.0016
.0023
.0031
.0041
.0055
.0073
.0096
.0125
.0162
.0207
.0262
.0329
.0409
.0505
.0618
.0749
.0901
.1075
.1271
.1492
.1736
.2005
.2296
.2611
.2946
.3300
.3669
.4052
.4443
.4840
.0003
.0004
.0006
.0008
.0011
.0016
.0022
.0030
.0040
.0054
.0071
.0094
.0122
.0158
.0202
.0256
.0322
.0401
.0495
.0606
.0735
.0885
.1056
.1251
.1469
.1711
.1977
.2266
.2578
.2912
.3264
.3632
.4013
.4404
.4801
.0003
.0004
.0006
.0008
.0011
.0015
.0021
.0029
.0039
.0052
.0069
.0091
.0119
.0154
.0197
.0250
.0314
.0392
.0485
.0594
.0721
.0869
.1038
.1230
.1446
.1685
.1949
.2236
.2546
.2877
.3228
.3594
.3974
.4364
.4761
.0003
.0004
.0005
.0008
.0011
.0015
.0021
.0028
.0038
.0051
.0068
.0089
.0116
.0150
.0192
.0244
.0307
.0384
.0475
.0582
.0708
.0853
.1020
.1210
.1423
.1660
.1922
.2206
.2514
.2843
.3192
.3557
.3936
.4325
.4721
.0003
.0004
.0005
.0007
.0010
.0014
.0020
.0027
.0037
.0049
.0066
.0087
.0113
.0146
.0188
.0239
.0301
.0375
.0465
.0571
.0694
.0838
.1003
.1190
.1401
.1635
.1894
.2177
.2483
.2810
.3156
.3520
.3897
.4286
.4681
.0002
.0003
.0005
.0007
.0010
.0014
.0019
.0026
.0036
.0048
.0064
.0084
.0110
.0143
.0183
.0233
.0294
.0367
.0455
.0559
.0681
.0823
.0985
.1170
.1379
.1611
.1867
.2148
.2451
.2776
.3121
.3483
.3859
.4247
.4641
12
2006 AP® STATISTICS FREE-RESPONSE QUESTIONS
Probability
Table entry for z is the
probability lying below z.
z
Table A
(Continued)
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
.5000
.5398
.5793
.6179
.6554
.6915
.7257
.7580
.7881
.8159
.8413
.8643
.8849
.9032
.9192
.9332
.9452
.9554
.9641
.9713
.9772
.9821
.9861
.9893
.9918
.9938
.9953
.9965
.9974
.9981
.9987
.9990
.9993
.9995
.9997
.5040
.5438
.5832
.6217
.6591
.6950
.7291
.7611
.7910
.8186
.8438
.8665
.8869
.9049
.9207
.9345
.9463
.9564
.9649
.9719
.9778
.9826
.9864
.9896
.9920
.9940
.9955
.9966
.9975
.9982
.9987
.9991
.9993
.9995
.9997
.5080
.5478
.5871
.6255
.6628
.6985
.7324
.7642
.7939
.8212
.8461
.8686
.8888
.9066
.9222
.9357
.9474
.9573
.9656
.9726
.9783
.9830
.9868
.9898
.9922
.9941
.9956
.9967
.9976
.9982
.9987
.9991
.9994
.9995
.9997
.5120
.5517
.5910
.6293
.6664
.7019
.7357
.7673
.7967
.8238
.8485
.8708
.8907
.9082
.9236
.9370
.9484
.9582
.9664
.9732
.9788
.9834
.9871
.9901
.9925
.9943
.9957
.9968
.9977
.9983
.9988
.9991
.9994
.9996
.9997
.5160
.5557
.5948
.6331
.6700
.7054
.7389
.7704
.7995
.8264
.8508
.8729
.8925
.9099
.9251
.9382
.9495
.9591
.9671
.9738
.9793
.9838
.9875
.9904
.9927
.9945
.9959
.9969
.9977
.9984
.9988
.9992
.9994
.9996
.9997
.5199
.5596
.5987
.6368
.6736
.7088
.7422
.7734
.8023
.8289
.8531
.8749
.8944
.9115
.9265
.9394
.9505
.9599
.9678
.9744
.9798
.9842
.9878
.9906
.9929
.9946
.9960
.9970
.9978
.9984
.9989
.9992
.9994
.9996
.9997
.5239
.5636
.6026
.6406
.6772
.7123
.7454
.7764
.8051
.8315
.8554
.8770
.8962
.9131
.9279
.9406
.9515
.9608
.9686
.9750
.9803
.9846
.9881
.9909
.9931
.9948
.9961
.9971
.9979
.9985
.9989
.9992
.9994
.9996
.9997
.5279
.5675
.6064
.6443
.6808
.7157
.7486
.7794
.8078
.8340
.8577
.8790
.8980
.9147
.9292
.9418
.9525
.9616
.9693
.9756
.9808
.9850
.9884
.9911
.9932
.9949
.9962
.9972
.9979
.9985
.9989
.9992
.9995
.9996
.9997
.5319
.5714
.6103
.6480
.6844
.7190
.7517
.7823
.8106
.8365
.8599
.8810
.8997
.9162
.9306
.9429
.9535
.9625
.9699
.9761
.9812
.9854
.9887
.9913
.9934
.9951
.9963
.9973
.9980
.9986
.9990
.9993
.9995
.9996
.9997
.5359
.5753
.6141
.6517
.6879
.7224
.7549
.7852
.8133
.8389
.8621
.8830
.9015
.9177
.9319
.9441
.9545
.9633
.9706
.9767
.9817
.9857
.9890
.9916
.9936
.9952
.9964
.9974
.9981
.9986
.9990
.9993
.9995
.9997
.9998
13
2006 AP® STATISTICS FREE-RESPONSE QUESTIONS
Table entry for p and
C is the point t* with
probability p lying
above it and
probability C lying
between −t * and t*.
Probability p
t*
Table B
t distribution critical values
Tail probability p
df
.25
.20
.15
.10
.05
.025
.02
.01
.005
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
80
100
1000
!
1.000
.816
.765
.741
.727
.718
.711
.706
.703
.700
.697
.695
.694
.692
.691
.690
.689
.688
.688
.687
.686
.686
.685
.685
.684
.684
.684
.683
.683
.683
.681
.679
.679
.678
.677
.675
.674
1.376
1.061
.978
.941
.920
.906
.896
.889
.883
.879
.876
.873
.870
.868
.866
.865
.863
.862
.861
.860
.859
.858
.858
.857
.856
.856
.855
.855
.854
.854
.851
.849
.848
.846
.845
.842
.841
1.963
1.386
1.250
1.190
1.156
1.134
1.119
1.108
1.100
1.093
1.088
1.083
1.079
1.076
1.074
1.071
1.069
1.067
1.066
1.064
1.063
1.061
1.060
1.059
1.058
1.058
1.057
1.056
1.055
1.055
1.050
1.047
1.045
1.043
1.042
1.037
1.036
3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325
1.323
1.321
1.319
1.318
1.316
1.315
1.314
1.313
1.311
1.310
1.303
1.299
1.296
1.292
1.290
1.282
1.282
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
1.711
1.708
1.706
1.703
1.701
1.699
1.697
1.684
1.676
1.671
1.664
1.660
1.646
1.645
12.71
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.080
2.074
2.069
2.064
2.060
2.056
2.052
2.048
2.045
2.042
2.021
2.009
2.000
1.990
1.984
1.962
1.960
15.89
4.849
3.482
2.999
2.757
2.612
2.517
2.449
2.398
2.359
2.328
2.303
2.282
2.264
2.249
2.235
2.224
2.214
2.205
2.197
2.189
2.183
2.177
2.172
2.167
2.162
2.158
2.154
2.150
2.147
2.123
2.109
2.099
2.088
2.081
2.056
2.054
31.82
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
2.718
2.681
2.650
2.624
2.602
2.583
2.567
2.552
2.539
2.528
2.518
2.508
2.500
2.492
2.485
2.479
2.473
2.467
2.462
2.457
2.423
2.403
2.390
2.374
2.364
2.330
2.326
63.66
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845
2.831
2.819
2.807
2.797
2.787
2.779
2.771
2.763
2.756
2.750
2.704
2.678
2.660
2.639
2.626
2.581
2.576
127.3
14.09
7.453
5.598
4.773
4.317
4.029
3.833
3.690
3.581
3.497
3.428
3.372
3.326
3.286
3.252
3.222
3.197
3.174
3.153
3.135
3.119
3.104
3.091
3.078
3.067
3.057
3.047
3.038
3.030
2.971
2.937
2.915
2.887
2.871
2.813
2.807
318.3
22.33
10.21
7.173
5.893
5.208
4.785
4.501
4.297
4.144
4.025
3.930
3.852
3.787
3.733
3.686
3.646
3.611
3.579
3.552
3.527
3.505
3.485
3.467
3.450
3.435
3.421
3.408
3.396
3.385
3.307
3.261
3.232
3.195
3.174
3.098
3.091
636.6
31.60
12.92
8.610
6.869
5.959
5.408
5.041
4.781
4.587
4.437
4.318
4.221
4.140
4.073
4.015
3.965
3.922
3.883
3.850
3.819
3.792
3.768
3.745
3.725
3.707
3.690
3.674
3.659
3.646
3.551
3.496
3.460
3.416
3.390
3.300
3.291
50%
60%
70%
80%
90%
95%
96%
98%
99%
99.5%
99.8%
99.9%
Confidence level C
14
.0025
.001
.0005
2006 AP® STATISTICS FREE-RESPONSE QUESTIONS
Probability p
Table entry for p is the point
( χ 2 ) with probability p lying
above it.
(χ2 )
Table C
χ 2 critical values
Tail probability p
df
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
80
100
.25
1.32
2.77
4.11
5.39
6.63
7.84
9.04
10.22
11.39
12.55
13.70
14.85
15.98
17.12
18.25
19.37
20.49
21.60
22.72
23.83
24.93
26.04
27.14
28.24
29.34
30.43
31.53
32.62
33.71
34.80
45.62
56.33
66.98
88.13
109.1
.20
1.64
3.22
4.64
5.99
7.29
8.56
9.80
11.03
12.24
13.44
14.63
15.81
16.98
18.15
19.31
20.47
21.61
22.76
23.90
25.04
26.17
27.30
28.43
29.55
30.68
31.79
32.91
34.03
35.14
36.25
47.27
58.16
68.97
90.41
111.7
.15
.10
2.07
3.79
5.32
6.74
8.12
9.45
10.75
12.03
13.29
14.53
15.77
16.99
18.20
19.41
20.60
21.79
22.98
24.16
25.33
26.50
27.66
28.82
29.98
31.13
32.28
33.43
34.57
35.71
36.85
37.99
49.24
60.35
71.34
93.11
114.7
2.71
4.61
6.25
7.78
9.24
10.64
12.02
13.36
14.68
15.99
17.28
18.55
19.81
21.06
22.31
23.54
24.77
25.99
27.20
28.41
29.62
30.81
32.01
33.20
34.38
35.56
36.74
37.92
39.09
40.26
51.81
63.17
74.40
96.58
118.5
.05
3.84
5.99
7.81
9.49
11.07
12.59
14.07
15.51
16.92
18.31
19.68
21.03
22.36
23.68
25.00
26.30
27.59
28.87
30.14
31.41
32.67
33.92
35.17
36.42
37.65
38.89
40.11
41.34
42.56
43.77
55.76
67.50
79.08
101.9
124.3
.025
5.02
7.38
9.35
11.14
12.83
14.45
16.01
17.53
19.02
20.48
21.92
23.34
24.74
26.12
27.49
28.85
30.19
31.53
32.85
34.17
35.48
36.78
38.08
39.36
40.65
41.92
43.19
44.46
45.72
46.98
59.34
71.42
83.30
106.6
129.6
15
.02
.01
.005
5.41
7.82
9.84
11.67
13.39
15.03
16.62
18.17
19.68
21.16
22.62
24.05
25.47
26.87
28.26
29.63
31.00
32.35
33.69
35.02
36.34
37.66
38.97
40.27
41.57
42.86
44.14
45.42
46.69
47.96
60.44
72.61
84.58
108.1
131.1
6.63
9.21
11.34
13.28
15.09
16.81
18.48
20.09
21.67
23.21
24.72
26.22
27.69
29.14
30.58
32.00
33.41
34.81
36.19
37.57
38.93
40.29
41.64
42.98
44.31
45.64
46.96
48.28
49.59
50.89
63.69
76.15
88.38
112.3
135.8
7.88
10.60
12.84
14.86
16.75
18.55
20.28
21.95
23.59
25.19
26.76
28.30
29.82
31.32
32.80
34.27
35.72
37.16
38.58
40.00
41.40
42.80
44.18
45.56
46.93
48.29
49.64
50.99
52.34
53.67
66.77
79.49
91.95
116.3
140.2
.0025
9.14
11.98
14.32
16.42
18.39
20.25
22.04
23.77
25.46
27.11
28.73
30.32
31.88
33.43
34.95
36.46
37.95
39.42
40.88
42.34
43.78
45.20
46.62
48.03
49.44
50.83
52.22
53.59
54.97
56.33
69.70
82.66
95.34
120.1
144.3
.001
.0005
10.83
13.82
16.27
18.47
20.51
22.46
24.32
26.12
27.88
29.59
31.26
32.91
34.53
36.12
37.70
39.25
40.79
42.31
43.82
45.31
46.80
48.27
49.73
51.18
52.62
54.05
55.48
56.89
58.30
59.70
73.40
86.66
99.61
124.8
149.4
12.12
15.20
17.73
20.00
22.11
24.10
26.02
27.87
29.67
31.42
33.14
34.82
36.48
38.11
39.72
41.31
42.88
44.43
45.97
47.50
49.01
50.51
52.00
53.48
54.95
56.41
57.86
59.30
60.73
62.16
76.09
89.56
102.7
128.3
153.2
AP® Statistics
2007 Free-Response Questions
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2007 AP® STATISTICS FREE-RESPONSE QUESTIONS
Formulas begin on page 3.
Questions begin on page 6.
Tables begin on page 12.
-2-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS
Formulas
(I)
Descriptive Statistics
∑ xi
n
x =
d
i
sx =
2
1
∑ xi − x
n −1
sp =
dn1 − 1is21 + dn2 − 1is22
dn1 − 1i + dn2 − 1i
y = b0 + b1 x
d
id
∑ xi − x y i − y
b1 =
d
∑ xi − x
i
i2
b0 = y − b1 x
r =
b1 = r
sb =
1
FG
H
x − x
1
∑ i
sx
n −1
IJ FG yi − y IJ
K H sy K
sy
sx
d
i2
d
i2
∑ yi − yi
n−2
∑ xi − x
-3-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS
(II)
Probability
P( A ∪ B) = P( A) + P( B) − P( A ∩ B)
P( A B ) =
P( A ∩ B )
P( B )
E ( X ) = μ x = ∑ xi pi
d
Var( X ) = s 2x = ∑ xi − μ x
i2 pi
If X has a binomial distribution with parameters n and p, then:
P( X = k ) =
FG nIJ p k (1 − p)n − k
H kK
μ x = np
sx =
μ
s
np(1 − p)
p
= p
p
=
p(1 − p)
n
If x is the mean of a random sample of size n from an infinite
population with mean μ and standard deviation s, then:
μx = μ
sx =
s
n
-4-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS
(III)
Inferential Statistics
Standardized test statistic:
statistic - parameter
standard deviation of statistic
Confidence interval: statistic ± (critical value ) ∑ (standard deviation of statistic)
Single-Sample
Standard Deviation
of Statistic
Statistic
σ
Sample Mean
n
p(1 − p)
n
Sample Proportion
Two-Sample
Standard Deviation
of Statistic
Statistic
Difference of
sample means
σ 12
n1
+
σ 22
n2
Special case when σ 1 = σ 2
σ
Difference of
sample proportions
1
1
+
n1 n2
p1 (1 − p1 )
p (1 − p2 )
+ 2
n1
n2
Special case when p1 = p2
b g
p 1− p
1
1
+
n1 n2
aobserved − expectedf
Chi-square test statistic = ∑
expected
-5-
2
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS
STATISTICS
SECTION II
Part A
Questions 1-5
Spend about 65 minutes on this part of the exam.
Percent of Section II grade—75
Directions: Show all your work. Indicate clearly the methods you use, because you will be graded on the
correctness of your methods as well as on the accuracy and completeness of your results and explanations.
1. The department of agriculture at a university was interested in determining whether a preservative was effective
in reducing discoloration in frozen strawberries. A sample of 50 ripe strawberries was prepared for freezing.
Then the sample was randomly divided into two groups of 25 strawberries each. Each strawberry was placed
into a small plastic bag.
The 25 bags in the control group were sealed. The preservative was added to the 25 bags containing strawberries
in the treatment group, and then those bags were sealed. All bags were stored at 0!C for a period of 6 months. At
the end of this time, after the strawberries were thawed, a technician rated each strawberry’s discoloration from
1 to 10, with a low score indicating little discoloration.
The dotplots below show the distributions of discoloration rating for the control and treatment groups.
(a) The standard deviation of ratings for the control group is 2.141. Explain how this value summarizes
variability in the control group.
(b) Based on the dotplots, comment on the effectiveness of the preservative in lowering the amount of
discoloration in strawberries. (No calculations are necessary.)
(c) Researchers at the university decided to calculate a 95 percent confidence interval for the difference in mean
discoloration rating between strawberries that were not treated with preservative and those that were treated
with preservative. The confidence interval they obtained was (0.16, 2.72). Assume that the conditions
necessary for the t-confidence interval are met.
Based on the confidence interval, comment on whether there would be a difference in the population mean
discoloration ratings for the treated and untreated strawberries.
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2007 AP® STATISTICS FREE-RESPONSE QUESTIONS
2. As dogs age, diminished joint and hip health may lead to joint pain and thus reduce a dog’s activity level. Such
a reduction in activity can lead to other health concerns such as weight gain and lethargy due to lack of exercise.
A study is to be conducted to see which of two dietary supplements, glucosamine or chondroitin, is more
effective in promoting joint and hip health and reducing the onset of canine osteoarthritis. Researchers will
randomly select a total of 300 dogs from ten different large veterinary practices around the country. All of the
dogs are more than 6 years old, and their owners have given consent to participate in the study. Changes in joint
and hip health will be evaluated after 6 months of treatment.
(a) What would be an advantage to adding a control group in the design of this study?
(b) Assuming a control group is added to the other two groups in the study, explain how you would assign the
300 dogs to these three groups for a completely randomized design.
(c) Rather than using a completely randomized design, one group of researchers proposes blocking on clinics,
and another group of researchers proposes blocking on breed of dog. How would you decide which one of
these two variables to use as a blocking variable?
3. Big Town Fisheries recently stocked a new lake in a city park with 2,000 fish of various sizes. The distribution
of the lengths of these fish is approximately normal.
(a) Big Town Fisheries claims that the mean length of the fish is 8 inches. If the claim is true, which of the
following would be more likely?
• A random sample of 15 fish having a mean length that is greater than 10 inches
or
• A random sample of 50 fish having a mean length that is greater than 10 inches
Justify your answer.
(b) Suppose the standard deviation of the sampling distribution of the sample mean for random samples of size
50 is 0.3 inch. If the mean length of the fish is 8 inches, use the normal distribution to compute the
probability that a random sample of 50 fish will have a mean length less than 7.5 inches.
(c) Suppose the distribution of fish lengths in this lake was nonnormal but had the same mean and standard
deviation. Would it still be appropriate to use the normal distribution to compute the probability in part (b) ?
Justify your answer.
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2007 AP® STATISTICS FREE-RESPONSE QUESTIONS
4. Investigators at the U.S. Department of Agriculture wished to compare methods of determining the level
of E. coli bacteria contamination in beef. Two different methods (A and B) of determining the level of
contamination were used on each of ten randomly selected specimens of a certain type of beef. The data
obtained, in millimicrobes/liter of ground beef, for each of the methods are shown in the table below.
Specimen
Method
1
2
3
4
5
6
7
8
9
10
A
22.7
23.6
24.0
27.1
27.4
27.8
34.4
35.2
40.4
46.8
B
23.0
23.1
23.7
26.5
26.6
27.1
33.2
35.0
40.5
47.8
Is there a significant difference in the mean amount of E. coli bacteria detected by the two methods for this
type of beef? Provide a statistical justification to support your answer.
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2007 AP® STATISTICS FREE-RESPONSE QUESTIONS
5. Researchers want to determine whether drivers are significantly more distracted while driving when using a
cell phone than when talking to a passenger in the car. In a study involving 48 people, 24 people were randomly
assigned to drive in a driving simulator while using a cell phone. The remaining 24 were assigned to drive in
the driving simulator while talking to a passenger in the simulator. Part of the driving simulation for both groups
involved asking drivers to exit the freeway at a particular exit. In the study, 7 of the 24 cell phone users missed
the exit, while 2 of the 24 talking to a passenger missed the exit.
(a) Would this study be classified as an experiment or an observational study? Provide an explanation to support
your answer.
(b) State the null and alternative hypotheses of interest to the researchers.
(c) One test of significance that you might consider using to answer the researchers’ question is a two-sample
z-test. State the conditions required for this test to be appropriate. Then comment on whether each condition
is met.
(d) Using an advanced statistical method for small samples to test the hypotheses in part (b), the researchers
report a p-value of 0.0683. Interpret, in everyday language, what this p-value measures in the context
of this study and state what conclusion should be made based on this p-value.
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2007 AP® STATISTICS FREE-RESPONSE QUESTIONS
STATISTICS
SECTION II
Part B
Question 6
Spend about 25 minutes on this part of the exam.
Percent of Section II grade—25
Directions: Show all your work. Indicate clearly the methods you use, because you will be graded on the
correctness of your methods as well as on the accuracy and completeness of your results and explanations.
6. A study was designed to explore subjects’ ability to judge the distance between two objects placed in a dimly lit
room. The researcher suspected that the subjects would generally overestimate the distance between the objects
in the room and that this overestimation would increase the farther apart the objects were.
The two objects were placed at random locations in the room before a subject estimated the distance (in feet)
between those two objects. After each subject estimated the distance, the locations of the objects were
rerandomized before the next subject viewed the room.
After data were collected for 40 subjects, two linear models were fit in an attempt to describe the relationship
between the subjects’ perceived distances (y) and the actual distance, in feet, between the two objects.
Model 1: yˆ = 0.238 + 1.080 " (actual distance)
The standard errors of the estimated coefficients for Model 1 are 0.260 and 0.118, respectively.
Model 2: yˆ = 1.102 " (actual distance)
The standard error of the estimated coefficient for Model 2 is 0.393.
(a) Provide an interpretation in context for the estimated slope in Model 1.
(b) Explain why the researcher might prefer Model 2 to Model 1 in this context.
(c) Using Model 2, test the researcher’s hypothesis that in dim light participants overestimate the distance, with
the overestimate increasing as the actual distance increases. (Assume appropriate conditions for inference
are met.)
The researchers also wanted to explore whether the performance on this task differed between subjects who wear
contact lenses and subjects who do not wear contact lenses. A new variable was created to indicate whether or
not a subject wears contact lenses. The data for this variable were coded numerically (1 = contact wearer,
0 = noncontact wearer), and this new variable, named “contact,” was included in the following model.
Model 3: yˆ = 1.05 " (actual distance) + 0.12 " (contact) " (actual distance )
The standard errors of the estimated coefficients for Model 3 are 0.357 and 0.032, respectively.
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-10-
GO ON TO THE NEXT PAGE.
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS
(d) Using Model 3, sketch the estimated regression model for contact wearers and the estimated regression
model for noncontact wearers on the grid below.
(e) In the context of this study, provide an interpretation of the estimated coefficients for Model 3.
STOP
END OF EXAM
© 2007 The College Board. All rights reserved.
Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents).
-11-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS
Probability
Table entry for z is the
probability lying below z.
Table A
z
Standard normal probabilities
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
– 3.4
– 3.3
– 3.2
– 3.1
– 3.0
– 2.9
– 2.8
– 2.7
– 2.6
– 2.5
– 2.4
– 2.3
– 2.2
– 2.1
– 2.0
– 1.9
– 1.8
– 1.7
– 1.6
– 1.5
– 1.4
– 1.3
– 1.2
– 1.1
– 1.0
– 0.9
– 0.8
– 0.7
– 0.6
– 0.5
– 0.4
– 0.3
– 0.2
– 0.1
– 0.0
.0003
.0005
.0007
.0010
.0013
.0019
.0026
.0035
.0047
.0062
.0082
.0107
.0139
.0179
.0228
.0287
.0359
.0446
.0548
.0668
.0808
.0968
.1151
.1357
.1587
.1841
.2119
.2420
.2743
.3085
.3446
.3821
.4207
.4602
.5000
.0003
.0005
.0007
.0009
.0013
.0018
.0025
.0034
.0045
.0060
.0080
.0104
.0136
.0174
.0222
.0281
.0351
.0436
.0537
.0655
.0793
.0951
.1131
.1335
.1562
.1814
.2090
.2389
.2709
.3050
.3409
.3783
.4168
.4562
.4960
.0003
.0005
.0006
.0009
.0013
.0018
.0024
.0033
.0044
.0059
.0078
.0102
.0132
.0170
.0217
.0274
.0344
.0427
.0526
.0643
.0778
.0934
.1112
.1314
.1539
.1788
.2061
.2358
.2676
.3015
.3372
.3745
.4129
.4522
.4920
.0003
.0004
.0006
.0009
.0012
.0017
.0023
.0032
.0043
.0057
.0075
.0099
.0129
.0166
.0212
.0268
.0336
.0418
.0516
.0630
.0764
.0918
.1093
.1292
.1515
.1762
.2033
.2327
.2643
.2981
.3336
.3707
.4090
.4483
.4880
.0003
.0004
.0006
.0008
.0012
.0016
.0023
.0031
.0041
.0055
.0073
.0096
.0125
.0162
.0207
.0262
.0329
.0409
.0505
.0618
.0749
.0901
.1075
.1271
.1492
.1736
.2005
.2296
.2611
.2946
.3300
.3669
.4052
.4443
.4840
.0003
.0004
.0006
.0008
.0011
.0016
.0022
.0030
.0040
.0054
.0071
.0094
.0122
.0158
.0202
.0256
.0322
.0401
.0495
.0606
.0735
.0885
.1056
.1251
.1469
.1711
.1977
.2266
.2578
.2912
.3264
.3632
.4013
.4404
.4801
.0003
.0004
.0006
.0008
.0011
.0015
.0021
.0029
.0039
.0052
.0069
.0091
.0119
.0154
.0197
.0250
.0314
.0392
.0485
.0594
.0721
.0869
.1038
.1230
.1446
.1685
.1949
.2236
.2546
.2877
.3228
.3594
.3974
.4364
.4761
.0003
.0004
.0005
.0008
.0011
.0015
.0021
.0028
.0038
.0051
.0068
.0089
.0116
.0150
.0192
.0244
.0307
.0384
.0475
.0582
.0708
.0853
.1020
.1210
.1423
.1660
.1922
.2206
.2514
.2843
.3192
.3557
.3936
.4325
.4721
.0003
.0004
.0005
.0007
.0010
.0014
.0020
.0027
.0037
.0049
.0066
.0087
.0113
.0146
.0188
.0239
.0301
.0375
.0465
.0571
.0694
.0838
.1003
.1190
.1401
.1635
.1894
.2177
.2483
.2810
.3156
.3520
.3897
.4286
.4681
.0002
.0003
.0005
.0007
.0010
.0014
.0019
.0026
.0036
.0048
.0064
.0084
.0110
.0143
.0183
.0233
.0294
.0367
.0455
.0559
.0681
.0823
.0985
.1170
.1379
.1611
.1867
.2148
.2451
.2776
.3121
.3483
.3859
.4247
.4641
-12-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS
Probability
Table entry for z is the
probability lying below z.
z
Table A
(Continued)
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
.5000
.5398
.5793
.6179
.6554
.6915
.7257
.7580
.7881
.8159
.8413
.8643
.8849
.9032
.9192
.9332
.9452
.9554
.9641
.9713
.9772
.9821
.9861
.9893
.9918
.9938
.9953
.9965
.9974
.9981
.9987
.9990
.9993
.9995
.9997
.5040
.5438
.5832
.6217
.6591
.6950
.7291
.7611
.7910
.8186
.8438
.8665
.8869
.9049
.9207
.9345
.9463
.9564
.9649
.9719
.9778
.9826
.9864
.9896
.9920
.9940
.9955
.9966
.9975
.9982
.9987
.9991
.9993
.9995
.9997
.5080
.5478
.5871
.6255
.6628
.6985
.7324
.7642
.7939
.8212
.8461
.8686
.8888
.9066
.9222
.9357
.9474
.9573
.9656
.9726
.9783
.9830
.9868
.9898
.9922
.9941
.9956
.9967
.9976
.9982
.9987
.9991
.9994
.9995
.9997
.5120
.5517
.5910
.6293
.6664
.7019
.7357
.7673
.7967
.8238
.8485
.8708
.8907
.9082
.9236
.9370
.9484
.9582
.9664
.9732
.9788
.9834
.9871
.9901
.9925
.9943
.9957
.9968
.9977
.9983
.9988
.9991
.9994
.9996
.9997
.5160
.5557
.5948
.6331
.6700
.7054
.7389
.7704
.7995
.8264
.8508
.8729
.8925
.9099
.9251
.9382
.9495
.9591
.9671
.9738
.9793
.9838
.9875
.9904
.9927
.9945
.9959
.9969
.9977
.9984
.9988
.9992
.9994
.9996
.9997
.5199
.5596
.5987
.6368
.6736
.7088
.7422
.7734
.8023
.8289
.8531
.8749
.8944
.9115
.9265
.9394
.9505
.9599
.9678
.9744
.9798
.9842
.9878
.9906
.9929
.9946
.9960
.9970
.9978
.9984
.9989
.9992
.9994
.9996
.9997
.5239
.5636
.6026
.6406
.6772
.7123
.7454
.7764
.8051
.8315
.8554
.8770
.8962
.9131
.9279
.9406
.9515
.9608
.9686
.9750
.9803
.9846
.9881
.9909
.9931
.9948
.9961
.9971
.9979
.9985
.9989
.9992
.9994
.9996
.9997
.5279
.5675
.6064
.6443
.6808
.7157
.7486
.7794
.8078
.8340
.8577
.8790
.8980
.9147
.9292
.9418
.9525
.9616
.9693
.9756
.9808
.9850
.9884
.9911
.9932
.9949
.9962
.9972
.9979
.9985
.9989
.9992
.9995
.9996
.9997
.5319
.5714
.6103
.6480
.6844
.7190
.7517
.7823
.8106
.8365
.8599
.8810
.8997
.9162
.9306
.9429
.9535
.9625
.9699
.9761
.9812
.9854
.9887
.9913
.9934
.9951
.9963
.9973
.9980
.9986
.9990
.9993
.9995
.9996
.9997
.5359
.5753
.6141
.6517
.6879
.7224
.7549
.7852
.8133
.8389
.8621
.8830
.9015
.9177
.9319
.9441
.9545
.9633
.9706
.9767
.9817
.9857
.9890
.9916
.9936
.9952
.9964
.9974
.9981
.9986
.9990
.9993
.9995
.9997
.9998
-13-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS
Table entry for p and
C is the point t* with
probability p lying
above it and
probability C lying
between −t * and t*.
Probability p
t*
Table B
t distribution critical values
Tail probability p
df
.25
.20
.15
.10
.05
.025
.02
.01
.005
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
80
100
1000
!
1.000
.816
.765
.741
.727
.718
.711
.706
.703
.700
.697
.695
.694
.692
.691
.690
.689
.688
.688
.687
.686
.686
.685
.685
.684
.684
.684
.683
.683
.683
.681
.679
.679
.678
.677
.675
.674
1.376
1.061
.978
.941
.920
.906
.896
.889
.883
.879
.876
.873
.870
.868
.866
.865
.863
.862
.861
.860
.859
.858
.858
.857
.856
.856
.855
.855
.854
.854
.851
.849
.848
.846
.845
.842
.841
1.963
1.386
1.250
1.190
1.156
1.134
1.119
1.108
1.100
1.093
1.088
1.083
1.079
1.076
1.074
1.071
1.069
1.067
1.066
1.064
1.063
1.061
1.060
1.059
1.058
1.058
1.057
1.056
1.055
1.055
1.050
1.047
1.045
1.043
1.042
1.037
1.036
3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325
1.323
1.321
1.319
1.318
1.316
1.315
1.314
1.313
1.311
1.310
1.303
1.299
1.296
1.292
1.290
1.282
1.282
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
1.711
1.708
1.706
1.703
1.701
1.699
1.697
1.684
1.676
1.671
1.664
1.660
1.646
1.645
12.71
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.080
2.074
2.069
2.064
2.060
2.056
2.052
2.048
2.045
2.042
2.021
2.009
2.000
1.990
1.984
1.962
1.960
15.89
4.849
3.482
2.999
2.757
2.612
2.517
2.449
2.398
2.359
2.328
2.303
2.282
2.264
2.249
2.235
2.224
2.214
2.205
2.197
2.189
2.183
2.177
2.172
2.167
2.162
2.158
2.154
2.150
2.147
2.123
2.109
2.099
2.088
2.081
2.056
2.054
31.82
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
2.718
2.681
2.650
2.624
2.602
2.583
2.567
2.552
2.539
2.528
2.518
2.508
2.500
2.492
2.485
2.479
2.473
2.467
2.462
2.457
2.423
2.403
2.390
2.374
2.364
2.330
2.326
63.66
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845
2.831
2.819
2.807
2.797
2.787
2.779
2.771
2.763
2.756
2.750
2.704
2.678
2.660
2.639
2.626
2.581
2.576
127.3
14.09
7.453
5.598
4.773
4.317
4.029
3.833
3.690
3.581
3.497
3.428
3.372
3.326
3.286
3.252
3.222
3.197
3.174
3.153
3.135
3.119
3.104
3.091
3.078
3.067
3.057
3.047
3.038
3.030
2.971
2.937
2.915
2.887
2.871
2.813
2.807
318.3
22.33
10.21
7.173
5.893
5.208
4.785
4.501
4.297
4.144
4.025
3.930
3.852
3.787
3.733
3.686
3.646
3.611
3.579
3.552
3.527
3.505
3.485
3.467
3.450
3.435
3.421
3.408
3.396
3.385
3.307
3.261
3.232
3.195
3.174
3.098
3.091
636.6
31.60
12.92
8.610
6.869
5.959
5.408
5.041
4.781
4.587
4.437
4.318
4.221
4.140
4.073
4.015
3.965
3.922
3.883
3.850
3.819
3.792
3.768
3.745
3.725
3.707
3.690
3.674
3.659
3.646
3.551
3.496
3.460
3.416
3.390
3.300
3.291
50%
60%
70%
80%
90%
95%
96%
98%
99%
99.5%
99.8%
99.9%
Confidence level C
-14-
.0025
.001
.0005
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS
Probability p
Table entry for p is the point
( χ 2 ) with probability p lying
above it.
(χ2 )
Table C
χ 2 critical values
Tail probability p
df
.25
.20
.15
.10
.05
.025
.02
.01
.005
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
80
100
1.32
2.77
4.11
5.39
6.63
7.84
9.04
10.22
11.39
12.55
13.70
14.85
15.98
17.12
18.25
19.37
20.49
21.60
22.72
23.83
24.93
26.04
27.14
28.24
29.34
30.43
31.53
32.62
33.71
34.80
45.62
56.33
66.98
88.13
109.1
1.64
3.22
4.64
5.99
7.29
8.56
9.80
11.03
12.24
13.44
14.63
15.81
16.98
18.15
19.31
20.47
21.61
22.76
23.90
25.04
26.17
27.30
28.43
29.55
30.68
31.79
32.91
34.03
35.14
36.25
47.27
58.16
68.97
90.41
111.7
2.07
3.79
5.32
6.74
8.12
9.45
10.75
12.03
13.29
14.53
15.77
16.99
18.20
19.41
20.60
21.79
22.98
24.16
25.33
26.50
27.66
28.82
29.98
31.13
32.28
33.43
34.57
35.71
36.85
37.99
49.24
60.35
71.34
93.11
114.7
2.71
4.61
6.25
7.78
9.24
10.64
12.02
13.36
14.68
15.99
17.28
18.55
19.81
21.06
22.31
23.54
24.77
25.99
27.20
28.41
29.62
30.81
32.01
33.20
34.38
35.56
36.74
37.92
39.09
40.26
51.81
63.17
74.40
96.58
118.5
3.84
5.99
7.81
9.49
11.07
12.59
14.07
15.51
16.92
18.31
19.68
21.03
22.36
23.68
25.00
26.30
27.59
28.87
30.14
31.41
32.67
33.92
35.17
36.42
37.65
38.89
40.11
41.34
42.56
43.77
55.76
67.50
79.08
101.9
124.3
5.02
7.38
9.35
11.14
12.83
14.45
16.01
17.53
19.02
20.48
21.92
23.34
24.74
26.12
27.49
28.85
30.19
31.53
32.85
34.17
35.48
36.78
38.08
39.36
40.65
41.92
43.19
44.46
45.72
46.98
59.34
71.42
83.30
106.6
129.6
5.41
7.82
9.84
11.67
13.39
15.03
16.62
18.17
19.68
21.16
22.62
24.05
25.47
26.87
28.26
29.63
31.00
32.35
33.69
35.02
36.34
37.66
38.97
40.27
41.57
42.86
44.14
45.42
46.69
47.96
60.44
72.61
84.58
108.1
131.1
6.63
9.21
11.34
13.28
15.09
16.81
18.48
20.09
21.67
23.21
24.72
26.22
27.69
29.14
30.58
32.00
33.41
34.81
36.19
37.57
38.93
40.29
41.64
42.98
44.31
45.64
46.96
48.28
49.59
50.89
63.69
76.15
88.38
112.3
135.8
7.88
10.60
12.84
14.86
16.75
18.55
20.28
21.95
23.59
25.19
26.76
28.30
29.82
31.32
32.80
34.27
35.72
37.16
38.58
40.00
41.40
42.80
44.18
45.56
46.93
48.29
49.64
50.99
52.34
53.67
66.77
79.49
91.95
116.3
140.2
-15-
.0025
9.14
11.98
14.32
16.42
18.39
20.25
22.04
23.77
25.46
27.11
28.73
30.32
31.88
33.43
34.95
36.46
37.95
39.42
40.88
42.34
43.78
45.20
46.62
48.03
49.44
50.83
52.22
53.59
54.97
56.33
69.70
82.66
95.34
120.1
144.3
.001
.0005
10.83
13.82
16.27
18.47
20.51
22.46
24.32
26.12
27.88
29.59
31.26
32.91
34.53
36.12
37.70
39.25
40.79
42.31
43.82
45.31
46.80
48.27
49.73
51.18
52.62
54.05
55.48
56.89
58.30
59.70
73.40
86.66
99.61
124.8
149.4
12.12
15.20
17.73
20.00
22.11
24.10
26.02
27.87
29.67
31.42
33.14
34.82
36.48
38.11
39.72
41.31
42.88
44.43
45.97
47.50
49.01
50.51
52.00
53.48
54.95
56.41
57.86
59.30
60.73
62.16
76.09
89.56
102.7
128.3
153.2
AP® Statistics
2007 Free-Response Questions
Form B
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2007 AP® STATISTICS FREE-RESPONSE QUESTIONS (Form B)
Formulas begin on page 3.
Questions begin on page 6.
Tables begin on page 13.
-2-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS (Form B)
Formulas
(I)
Descriptive Statistics
 xi
n
x =
(
)
sx =
2
1
 xi - x
n -1
sp =
(n1 - 1)s21 + (n2 - 1)s22
(n1 - 1) + (n2 - 1)
ŷ = b0 + b1x
(
)(
 xi - x yi - y
b1 =
(
 xi - x
)2
)
b0 = y - b1x
r =
Êx 1
ÂÁ i
n - 1 Ë sx
b1 = r
sb =
1
x ˆ Ê yi - y ˆ
˜¯ Á s ˜
Ë y ¯
sy
sx
(
)2
(
)2
 yi - yˆi
n-2
 xi - x
-3-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS (Form B)
(II)
Probability
P ( A » B ) = P ( A) + P ( B ) - P ( A « B )
P ( A B) =
P ( A « B)
P ( B)
E ( X ) = µ x = Â xi pi
(
Var( X ) = s 2x = Â xi - µ x
)
2
pi
If X has a binomial distribution with parameters n and p, then:
Ê nˆ
P ( X = k ) = Á ˜ p k (1 - p)n - k
Ë k¯
µ x = np
s x = np(1 - p)
µ pˆ = p
p(1 - p)
n
s pˆ =
If x is the mean of a random sample of size n from an infinite
population with mean µ and standard deviation s , then:
µx = µ
sx =
s
n
-4-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS (Form B)
(III)
Inferential Statistics
Standardized test statistic:
statistic - parameter
standard deviation of statistic
Confidence interval: statistic ± (critical value ) ∑ (standard deviation of statistic )
Single-Sample
Sample Mean
Standard Deviation
of Statistic
s
n
Sample Proportion
p(1 - p)
n
Statistic
Two-Sample
Standard Deviation
of Statistic
Statistic
Difference of
sample means
s12 s 22
+
n1
n2
Special case when s1 = s 2
s
Difference of
sample proportions
1
1
+
n1 n2
p1 (1 - p1 ) p2 (1 - p2 )
+
n1
n2
Special case when p1 = p2
p (1 - p )
Chi-square test statistic =
Â
-5-
1
1
+
n1 n2
(observed - expected )2
expected
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS (Form B)
STATISTICS
SECTION II
Part A
Questions 1-5
Spend about 65 minutes on this part of the exam.
Percent of Section II grade— 75
Directions: Show all your work. Indicate clearly the methods you use, because you will be graded on the
correctness of your methods as well as on the accuracy and completeness of your results and explanations.
1. The Better Business Council of a large city has concluded that students in the city’s schools are not learning
enough about economics to function in the modern world. These findings were based on test results from a
random sample of 20 twelfth-grade students who completed a 46-question multiple-choice test on basic
economic concepts. The data set below shows the number of questions that each of the 20 students in the
sample answered correctly.
12
19
16
36
18
19
17
13
18
43
33
8
41
16
44
14
38
10
35
9
(a) Display these data in a stemplot.
(b) Use your stemplot from part (a) to describe the main features of this score distribution.
(c) Why would it be misleading to report only a measure of center for this score distribution?
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-6-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS (Form B)
2. The graph below displays the relative frequency distribution for X, the total number of dogs and cats owned
per household, for the households in a large suburban area. For instance, 14 percent of the households own 2 of
these pets.
(a) According to a local law, each household in this area is prohibited from owning more than 3 of these pets. If
a household in this area is selected at random, what is the probability that the selected household will be in
violation of this law? Show your work.
(b) If 10 households in this area are selected at random, what is the probability that exactly 2 of them will be in
violation of this law? Show your work.
(c) The mean and standard deviation of X are 1.65 and 1.851, respectively. Suppose 150 households in this area
are to be selected at random and X , the mean number of dogs and cats per household, is to be computed.
Describe the sampling distribution of X , including its shape, center, and spread.
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-7-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS (Form B)
3. The United States Department of Energy is conducting an experiment to compare the heat gain in houses using
two different types of windows, A and B. Six windows of each type are available for the experiment. The
Department has constructed a house with twelve windows as shown on the floor plan below.
In the interior of the house, each window is surrounded by a window box to capture and measure the amount of
heat coming in through that window and to isolate the heat gain for each window.
(a) A randomized block experiment will be used to compare the heat gain for the two types (A and B) of
windows. How would you group the window boxes into blocks? (Clearly indicate your blocks using the
window box numbers.) Justify your choice of blocks.
(b) For the design in part (a), describe how you would assign window types (A and B) to the numbered window
boxes.
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-8-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS (Form B)
4. Each of 25 adult women was asked to provide her own height (y), in inches, and the height (x), in inches, of her
father. The scatterplot below displays the results. Only 22 of the 25 pairs are distinguishable because some of the
( x, y ) pairs were the same. The equation of the least squares regression line is yˆ = 35.1 + 0.427 x.
(a) Draw the least squares regression line on the scatterplot above.
(b) One father’s height was x = 67 inches and his daughter’s height was y = 61 inches. Circle the point on the
scatterplot above that represents this pair and draw the segment on the scatterplot that corresponds to the
residual for it. Give a numerical value for the residual.
(c) Suppose the point x = 84 , y = 71 is added to the data set. Would the slope of the least squares regression
line increase, decrease, or remain about the same? Explain.
(Note: No calculations are necessary to answer this question.)
Would the correlation increase, decrease, or remain about the same? Explain.
(Note: No calculations are necessary to answer this question.)
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-9-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS (Form B)
5. A serum cholesterol level above 250 milligrams per deciliter (mg/dl) of blood is a risk factor for cardiovascular
disease in humans. At a medical center in St. Louis, a study to test the effectiveness of a new cholesterollowering drug was conducted. One hundred people with cholesterol levels between 250 mg/dl and 300 mg/dl
were available for this study. Fifty people were assigned at random to each of two treatment groups. One group
received the standard cholesterol-lowering medication and the other group received the new drug. After taking
the drug for three weeks, the 50 subjects who received the standard treatment had a mean decrease in cholesterol
level of 10 mg/dl with a standard deviation of 8 mg/dl, and the 50 subjects who received the new drug had a
mean decrease of 18 mg/dl with a standard deviation of 12 mg/dl.
Does the new drug appear to be more effective than the standard treatment in lowering mean cholesterol level?
Give appropriate statistical evidence to support your conclusion.
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GO ON TO THE NEXT PAGE.
-10-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS (Form B)
STATISTICS
SECTION II
Part B
Question 6
Spend about 25 minutes on this part of the exam.
Percent of Section II grade— 25
Directions: Show all your work. Indicate clearly the methods you use, because you will be graded on the
correctness of your methods as well as on the accuracy and completeness of your results and explanations.
6. Scientists interested in preserving natural habitats and minimizing the possible extinction of certain bird species
conducted a study to determine if it is better for conservation groups to purchase a few large nature preserves or
many small preserves in order to meet these goals.
The scientists studied 13 randomly selected islands of different sizes to determine the risk of extinction for bird
species. Islands are thought to be a good imitation of what would happen in a nature preserve because of their
isolation. If a species lived on only one island, it was considered to be at risk. Scientists have determined that
whether or not one species becomes extinct is independent of whether or not another species becomes extinct.
In 1990 scientists counted the number of at-risk species on each of the selected islands. They returned to each of
these islands in the year 2000 to see whether the species still existed on the islands. Species that were present in
1990 but absent in 2000 were considered extinct. Data collected by the scientists are given in the table below.
Island
Area
(in sq km)
Species at Risk
in 1990
Species Extinct
by 2000
Proportion
Extinct
1
46
75
8
0.11
2
36
67
3
0.04
3
31
66
8
0.12
4
9
51
8
0.16
5
5
28
5
0.18
6
5
20
6
0.30
7
4
43
10
0.23
8
4
31
5
0.16
9
3
28
7
0.25
10
2
32
8
0.25
11
1
30
8
0.27
12
1
20
4
0.20
13
1
16
5
0.31
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GO ON TO THE NEXT PAGE.
-11-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS (Form B)
(a) One scientist involved in the study believes that large islands (those with areas greater than 25 square
kilometers) are more effective than small islands (those with areas of no more than 25 square kilometers)
for protecting at-risk species. The scientist noted that for this study, a total of 19 of the 208 species on the
large islands became extinct, whereas a total of 66 of the 299 species on the small islands became extinct.
Assume that the probability of extinction is the same for all at-risk species on large islands and the same
for all at-risk species on small islands. Do these data support the scientist's belief? Give appropriate
statistical justification for your answer.
(b) Another scientist who worked on this study thinks that the proportion of species that become extinct is
more directly related to the size of the islands than simply to whether the islands are grouped as large or
small. This scientist investigated the relationship between the proportion of extinct birds and the area, in
square kilometers, of islands. A least squares analysis was conducted on the proportion extinct and ln(area).
The regression analysis output, the scatterplot, and the residual plot are shown below.
Predictor
Constant
ln(area)
S = 0.02863
Coef
0.28996
-0.05323
StDev
0.01269
0.00618
T
22.85
-8.61
P
0.000
0.000
R-Sq = 87.1%
Estimate the slope of the least squares regression line using a 95 percent confidence interval. Interpret your
answer in the context of this situation.
(c) In part (a), the scientist assumed that the probability of a species becoming extinct is the same for each of the
large islands. Similarly, the scientist assumed that the probability is the same for each of the small islands.
Based on your answer in part (b), do you think this is a reasonable assumption? Explain.
(d) A conservation group with a long-term goal of preserving species believes that all at-risk species will
disappear whenever land inhabited by those species is developed. It has an opportunity to purchase land in
an area about to be developed. The group has a choice of creating one large nature preserve with an area
of 45 square kilometers and containing 70 at-risk species, or 5 small nature preserves, each with an area
of 3 square kilometers and each containing 16 at-risk species unique to that preserve. Which choice would
you recommend and why?
STOP
END OF EXAM
© 2007 The College Board. All rights reserved.
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-12-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS (Form B)
Probability
Table entry for z is the
probability lying below z.
Table A
z
Standard normal probabilities
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
– 3.4
– 3.3
– 3.2
– 3.1
– 3.0
– 2.9
– 2.8
– 2.7
– 2.6
– 2.5
– 2.4
– 2.3
– 2.2
– 2.1
– 2.0
– 1.9
– 1.8
– 1.7
– 1.6
– 1.5
– 1.4
– 1.3
– 1.2
– 1.1
– 1.0
– 0.9
– 0.8
– 0.7
– 0.6
– 0.5
– 0.4
– 0.3
– 0.2
– 0.1
– 0.0
.0003
.0005
.0007
.0010
.0013
.0019
.0026
.0035
.0047
.0062
.0082
.0107
.0139
.0179
.0228
.0287
.0359
.0446
.0548
.0668
.0808
.0968
.1151
.1357
.1587
.1841
.2119
.2420
.2743
.3085
.3446
.3821
.4207
.4602
.5000
.0003
.0005
.0007
.0009
.0013
.0018
.0025
.0034
.0045
.0060
.0080
.0104
.0136
.0174
.0222
.0281
.0351
.0436
.0537
.0655
.0793
.0951
.1131
.1335
.1562
.1814
.2090
.2389
.2709
.3050
.3409
.3783
.4168
.4562
.4960
.0003
.0005
.0006
.0009
.0013
.0018
.0024
.0033
.0044
.0059
.0078
.0102
.0132
.0170
.0217
.0274
.0344
.0427
.0526
.0643
.0778
.0934
.1112
.1314
.1539
.1788
.2061
.2358
.2676
.3015
.3372
.3745
.4129
.4522
.4920
.0003
.0004
.0006
.0009
.0012
.0017
.0023
.0032
.0043
.0057
.0075
.0099
.0129
.0166
.0212
.0268
.0336
.0418
.0516
.0630
.0764
.0918
.1093
.1292
.1515
.1762
.2033
.2327
.2643
.2981
.3336
.3707
.4090
.4483
.4880
.0003
.0004
.0006
.0008
.0012
.0016
.0023
.0031
.0041
.0055
.0073
.0096
.0125
.0162
.0207
.0262
.0329
.0409
.0505
.0618
.0749
.0901
.1075
.1271
.1492
.1736
.2005
.2296
.2611
.2946
.3300
.3669
.4052
.4443
.4840
.0003
.0004
.0006
.0008
.0011
.0016
.0022
.0030
.0040
.0054
.0071
.0094
.0122
.0158
.0202
.0256
.0322
.0401
.0495
.0606
.0735
.0885
.1056
.1251
.1469
.1711
.1977
.2266
.2578
.2912
.3264
.3632
.4013
.4404
.4801
.0003
.0004
.0006
.0008
.0011
.0015
.0021
.0029
.0039
.0052
.0069
.0091
.0119
.0154
.0197
.0250
.0314
.0392
.0485
.0594
.0721
.0869
.1038
.1230
.1446
.1685
.1949
.2236
.2546
.2877
.3228
.3594
.3974
.4364
.4761
.0003
.0004
.0005
.0008
.0011
.0015
.0021
.0028
.0038
.0051
.0068
.0089
.0116
.0150
.0192
.0244
.0307
.0384
.0475
.0582
.0708
.0853
.1020
.1210
.1423
.1660
.1922
.2206
.2514
.2843
.3192
.3557
.3936
.4325
.4721
.0003
.0004
.0005
.0007
.0010
.0014
.0020
.0027
.0037
.0049
.0066
.0087
.0113
.0146
.0188
.0239
.0301
.0375
.0465
.0571
.0694
.0838
.1003
.1190
.1401
.1635
.1894
.2177
.2483
.2810
.3156
.3520
.3897
.4286
.4681
.0002
.0003
.0005
.0007
.0010
.0014
.0019
.0026
.0036
.0048
.0064
.0084
.0110
.0143
.0183
.0233
.0294
.0367
.0455
.0559
.0681
.0823
.0985
.1170
.1379
.1611
.1867
.2148
.2451
.2776
.3121
.3483
.3859
.4247
.4641
-13-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS (Form B)
Probability
Table entry for z is the
probability lying below z.
z
Table A
(Continued)
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
.5000
.5398
.5793
.6179
.6554
.6915
.7257
.7580
.7881
.8159
.8413
.8643
.8849
.9032
.9192
.9332
.9452
.9554
.9641
.9713
.9772
.9821
.9861
.9893
.9918
.9938
.9953
.9965
.9974
.9981
.9987
.9990
.9993
.9995
.9997
.5040
.5438
.5832
.6217
.6591
.6950
.7291
.7611
.7910
.8186
.8438
.8665
.8869
.9049
.9207
.9345
.9463
.9564
.9649
.9719
.9778
.9826
.9864
.9896
.9920
.9940
.9955
.9966
.9975
.9982
.9987
.9991
.9993
.9995
.9997
.5080
.5478
.5871
.6255
.6628
.6985
.7324
.7642
.7939
.8212
.8461
.8686
.8888
.9066
.9222
.9357
.9474
.9573
.9656
.9726
.9783
.9830
.9868
.9898
.9922
.9941
.9956
.9967
.9976
.9982
.9987
.9991
.9994
.9995
.9997
.5120
.5517
.5910
.6293
.6664
.7019
.7357
.7673
.7967
.8238
.8485
.8708
.8907
.9082
.9236
.9370
.9484
.9582
.9664
.9732
.9788
.9834
.9871
.9901
.9925
.9943
.9957
.9968
.9977
.9983
.9988
.9991
.9994
.9996
.9997
.5160
.5557
.5948
.6331
.6700
.7054
.7389
.7704
.7995
.8264
.8508
.8729
.8925
.9099
.9251
.9382
.9495
.9591
.9671
.9738
.9793
.9838
.9875
.9904
.9927
.9945
.9959
.9969
.9977
.9984
.9988
.9992
.9994
.9996
.9997
.5199
.5596
.5987
.6368
.6736
.7088
.7422
.7734
.8023
.8289
.8531
.8749
.8944
.9115
.9265
.9394
.9505
.9599
.9678
.9744
.9798
.9842
.9878
.9906
.9929
.9946
.9960
.9970
.9978
.9984
.9989
.9992
.9994
.9996
.9997
.5239
.5636
.6026
.6406
.6772
.7123
.7454
.7764
.8051
.8315
.8554
.8770
.8962
.9131
.9279
.9406
.9515
.9608
.9686
.9750
.9803
.9846
.9881
.9909
.9931
.9948
.9961
.9971
.9979
.9985
.9989
.9992
.9994
.9996
.9997
.5279
.5675
.6064
.6443
.6808
.7157
.7486
.7794
.8078
.8340
.8577
.8790
.8980
.9147
.9292
.9418
.9525
.9616
.9693
.9756
.9808
.9850
.9884
.9911
.9932
.9949
.9962
.9972
.9979
.9985
.9989
.9992
.9995
.9996
.9997
.5319
.5714
.6103
.6480
.6844
.7190
.7517
.7823
.8106
.8365
.8599
.8810
.8997
.9162
.9306
.9429
.9535
.9625
.9699
.9761
.9812
.9854
.9887
.9913
.9934
.9951
.9963
.9973
.9980
.9986
.9990
.9993
.9995
.9996
.9997
.5359
.5753
.6141
.6517
.6879
.7224
.7549
.7852
.8133
.8389
.8621
.8830
.9015
.9177
.9319
.9441
.9545
.9633
.9706
.9767
.9817
.9857
.9890
.9916
.9936
.9952
.9964
.9974
.9981
.9986
.9990
.9993
.9995
.9997
.9998
-14-
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS (Form B)
Table entry for p and
C is the point t* with
probability p lying
above it and
probability C lying
between −t * and t*.
Probability p
t*
Table B
t distribution critical values
Tail probability p
df
.25
.20
.15
.10
.05
.025
.02
.01
.005
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
80
100
1000
!
1.000
.816
.765
.741
.727
.718
.711
.706
.703
.700
.697
.695
.694
.692
.691
.690
.689
.688
.688
.687
.686
.686
.685
.685
.684
.684
.684
.683
.683
.683
.681
.679
.679
.678
.677
.675
.674
1.376
1.061
.978
.941
.920
.906
.896
.889
.883
.879
.876
.873
.870
.868
.866
.865
.863
.862
.861
.860
.859
.858
.858
.857
.856
.856
.855
.855
.854
.854
.851
.849
.848
.846
.845
.842
.841
1.963
1.386
1.250
1.190
1.156
1.134
1.119
1.108
1.100
1.093
1.088
1.083
1.079
1.076
1.074
1.071
1.069
1.067
1.066
1.064
1.063
1.061
1.060
1.059
1.058
1.058
1.057
1.056
1.055
1.055
1.050
1.047
1.045
1.043
1.042
1.037
1.036
3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325
1.323
1.321
1.319
1.318
1.316
1.315
1.314
1.313
1.311
1.310
1.303
1.299
1.296
1.292
1.290
1.282
1.282
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
1.711
1.708
1.706
1.703
1.701
1.699
1.697
1.684
1.676
1.671
1.664
1.660
1.646
1.645
12.71
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.080
2.074
2.069
2.064
2.060
2.056
2.052
2.048
2.045
2.042
2.021
2.009
2.000
1.990
1.984
1.962
1.960
15.89
4.849
3.482
2.999
2.757
2.612
2.517
2.449
2.398
2.359
2.328
2.303
2.282
2.264
2.249
2.235
2.224
2.214
2.205
2.197
2.189
2.183
2.177
2.172
2.167
2.162
2.158
2.154
2.150
2.147
2.123
2.109
2.099
2.088
2.081
2.056
2.054
31.82
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
2.718
2.681
2.650
2.624
2.602
2.583
2.567
2.552
2.539
2.528
2.518
2.508
2.500
2.492
2.485
2.479
2.473
2.467
2.462
2.457
2.423
2.403
2.390
2.374
2.364
2.330
2.326
63.66
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845
2.831
2.819
2.807
2.797
2.787
2.779
2.771
2.763
2.756
2.750
2.704
2.678
2.660
2.639
2.626
2.581
2.576
127.3
14.09
7.453
5.598
4.773
4.317
4.029
3.833
3.690
3.581
3.497
3.428
3.372
3.326
3.286
3.252
3.222
3.197
3.174
3.153
3.135
3.119
3.104
3.091
3.078
3.067
3.057
3.047
3.038
3.030
2.971
2.937
2.915
2.887
2.871
2.813
2.807
318.3
22.33
10.21
7.173
5.893
5.208
4.785
4.501
4.297
4.144
4.025
3.930
3.852
3.787
3.733
3.686
3.646
3.611
3.579
3.552
3.527
3.505
3.485
3.467
3.450
3.435
3.421
3.408
3.396
3.385
3.307
3.261
3.232
3.195
3.174
3.098
3.091
636.6
31.60
12.92
8.610
6.869
5.959
5.408
5.041
4.781
4.587
4.437
4.318
4.221
4.140
4.073
4.015
3.965
3.922
3.883
3.850
3.819
3.792
3.768
3.745
3.725
3.707
3.690
3.674
3.659
3.646
3.551
3.496
3.460
3.416
3.390
3.300
3.291
50%
60%
70%
80%
90%
95%
96%
98%
99%
99.5%
99.8%
99.9%
Confidence level C
-15-
.0025
.001
.0005
2007 AP® STATISTICS FREE-RESPONSE QUESTIONS (Form B)
Probability p
Table entry for p is the point
( χ 2 ) with probability p lying
above it.
(χ2 )
Table C
c 2 critical values
Tail probability p
df
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
80
100
.25
1.32
2.77
4.11
5.39
6.63
7.84
9.04
10.22
11.39
12.55
13.70
14.85
15.98
17.12
18.25
19.37
20.49
21.60
22.72
23.83
24.93
26.04
27.14
28.24
29.34
30.43
31.53
32.62
33.71
34.80
45.62
56.33
66.98
88.13
109.1
.20
1.64
3.22
4.64
5.99
7.29
8.56
9.80
11.03
12.24
13.44
14.63
15.81
16.98
18.15
19.31
20.47
21.61
22.76
23.90
25.04
26.17
27.30
28.43
29.55
30.68
31.79
32.91
34.03
35.14
36.25
47.27
58.16
68.97
90.41
111.7
.15
.10
2.07
3.79
5.32
6.74
8.12
9.45
10.75
12.03
13.29
14.53
15.77
16.99
18.20
19.41
20.60
21.79
22.98
24.16
25.33
26.50
27.66
28.82
29.98
31.13
32.28
33.43
34.57
35.71
36.85
37.99
49.24
60.35
71.34
93.11
114.7
2.71
4.61
6.25
7.78
9.24
10.64
12.02
13.36
14.68
15.99
17.28
18.55
19.81
21.06
22.31
23.54
24.77
25.99
27.20
28.41
29.62
30.81
32.01
33.20
34.38
35.56
36.74
37.92
39.09
40.26
51.81
63.17
74.40
96.58
118.5
.05
3.84
5.99
7.81
9.49
11.07
12.59
14.07
15.51
16.92
18.31
19.68
21.03
22.36
23.68
25.00
26.30
27.59
28.87
30.14
31.41
32.67
33.92
35.17
36.42
37.65
38.89
40.11
41.34
42.56
43.77
55.76
67.50
79.08
101.9
124.3
.025
5.02
7.38
9.35
11.14
12.83
14.45
16.01
17.53
19.02
20.48
21.92
23.34
24.74
26.12
27.49
28.85
30.19
31.53
32.85
34.17
35.48
36.78
38.08
39.36
40.65
41.92
43.19
44.46
45.72
46.98
59.34
71.42
83.30
106.6
129.6
-16-
.02
.01
.005
5.41
7.82
9.84
11.67
13.39
15.03
16.62
18.17
19.68
21.16
22.62
24.05
25.47
26.87
28.26
29.63
31.00
32.35
33.69
35.02
36.34
37.66
38.97
40.27
41.57
42.86
44.14
45.42
46.69
47.96
60.44
72.61
84.58
108.1
131.1
6.63
9.21
11.34
13.28
15.09
16.81
18.48
20.09
21.67
23.21
24.72
26.22
27.69
29.14
30.58
32.00
33.41
34.81
36.19
37.57
38.93
40.29
41.64
42.98
44.31
45.64
46.96
48.28
49.59
50.89
63.69
76.15
88.38
112.3
135.8
7.88
10.60
12.84
14.86
16.75
18.55
20.28
21.95
23.59
25.19
26.76
28.30
29.82
31.32
32.80
34.27
35.72
37.16
38.58
40.00
41.40
42.80
44.18
45.56
46.93
48.29
49.64
50.99
52.34
53.67
66.77
79.49
91.95
116.3
140.2
.0025
.001
.0005
9.14
11.98
14.32
16.42
18.39
20.25
22.04
23.77
25.46
27.11
28.73
30.32
31.88
33.43
34.95
36.46
37.95
39.42
40.88
42.34
43.78
45.20
46.62
48.03
49.44
50.83
52.22
53.59
54.97
56.33
69.70
82.66
95.34
120.1
144.3
10.83
13.82
16.27
18.47
20.51
22.46
24.32
26.12
27.88
29.59
31.26
32.91
34.53
36.12
37.70
39.25
40.79
42.31
43.82
45.31
46.80
48.27
49.73
51.18
52.62
54.05
55.48
56.89
58.30
59.70
73.40
86.66
99.61
124.8
149.4
12.12
15.20
17.73
20.00
22.11
24.10
26.02
27.87
29.67
31.42
33.14
34.82
36.48
38.11
39.72
41.31
42.88
44.43
45.97
47.50
49.01
50.51
52.00
53.48
54.95
56.41
57.86
59.30
60.73
62.16
76.09
89.56
102.7
128.3
153.2
AP® Statistics
2009 Free-Response Questions
The College Board
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2009 AP® STATISTICS FREE-RESPONSE QUESTIONS
Formulas begin on page 3.
Questions begin on page 6.
Tables begin on page 11.
-2-
2009 AP® STATISTICS FREE-RESPONSE QUESTIONS
Formulas
(I)
Descriptive Statistics
 xi
n
x =
(
)
sx =
2
1
 xi - x
n -1
sp =
(n1 - 1)s21 + (n2 - 1)s22
(n1 - 1) + (n2 - 1)
ŷ = b0 + b1 x
(
)(
 xi - x yi - y
b1 =
(
 xi - x
)2
)
b0 = y - b1 x
r =
Êx 1
ÂÁ i
n - 1 Ë sx
b1 = r
sy
sx
(
 yi - yˆi
sb =
1
x ˆ Ê yi - y ˆ
˜Á s ˜
¯Ë y ¯
(
n-2
 xi - x
)2
)2
-3-
2009 AP® STATISTICS FREE-RESPONSE QUESTIONS
(II)
Probability
P ( A » B ) = P ( A) + P ( B ) - P ( A « B )
P ( A B) =
P ( A « B)
P ( B)
E ( X ) = μ x = Â xi pi
(
Var( X ) = s 2x = Â xi - μ x
)
2
pi
If X has a binomial distribution with parameters n and p, then:
Ê nˆ
P ( X = k ) = Á ˜ p k (1 - p)n - k
Ë k¯
μ x = np
s x = np(1 - p)
μ pˆ = p
p(1 - p)
n
s pˆ =
If x is the mean of a random sample of size n from an infinite
population with mean μ and standard deviation s , then:
μx = μ
sx =
s
n
-4-
2009 AP® STATISTICS FREE-RESPONSE QUESTIONS
(III)
Inferential Statistics
Standardized test statistic:
statistic - parameter
standard deviation of statistic
Confidence interval: statistic ± (critical value ) ∑ (standard deviation of statistic )
Single-Sample
Sample Mean
Standard Deviation
of Statistic
s
n
Sample Proportion
p(1 - p)
n
Statistic
Two-Sample
Standard Deviation
of Statistic
Statistic
Difference of
sample means
s12 s 22
+
n1
n2
Special case when s1 = s 2
s
Difference of
sample proportions
1
1
+
n1 n2
p1 (1 - p1 ) p2 (1 - p2 )
+
n1
n2
Special case when p1 = p2
p (1 - p )
Chi-square test statistic =
Â
-5-
1
1
+
n1 n2
(observed - expected )2
expected
2009 AP® STATISTICS FREE-RESPONSE QUESTIONS
STATISTICS
SECTION II
Part A
Questions 1-5
Spend about 65 minutes on this part of the exam.
Percent of Section II score—75
Directions: Show all your work. Indicate clearly the methods you use, because you will be graded on the
correctness of your methods as well as on the accuracy and completeness of your results and explanations.
1. A simple random sample of 100 high school seniors was selected from a large school district. The gender of each
student was recorded, and each student was asked the following questions.
1. Have you ever had a part-time job?
2. If you answered yes to the previous question, was your part-time job in the summer only?
The responses are summarized in the table below.
Gender
Job Experience
Male
Female
Total
Never had a part-time job
21
31
52
Had a part-time job during summer only
15
13
28
Had a part-time job but not only during summer
12
8
20
Total
48
52
100
(a) On the grid below, construct a graphical display that represents the association between gender and job
experience for the students in the sample.
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-6-
2009 AP® STATISTICS FREE-RESPONSE QUESTIONS
(b) Write a few sentences summarizing what the display in part (a) reveals about the association between gender
and job experience for the students in the sample.
(c) Which test of significance should be used to test if there is an association between gender and job experience
for the population of high school seniors in the district?
State the null and alternative hypotheses for the test, but do not perform the test.
2. A tire manufacturer designed a new tread pattern for its all-weather tires. Repeated tests were conducted on cars
of approximately the same weight traveling at 60 miles per hour. The tests showed that the new tread pattern
enables the cars to stop completely in an average distance of 125 feet with a standard deviation of 6.5 feet and
that the stopping distances are approximately normally distributed.
(a) What is the 70th percentile of the distribution of stopping distances?
(b) What is the probability that at least 2 cars out of 5 randomly selected cars in the study will stop in a distance
that is greater than the distance calculated in part (a) ?
(c) What is the probability that a randomly selected sample of 5 cars in the study will have a mean stopping
distance of at least 130 feet?
3. Before beginning a unit on frog anatomy, a seventh-grade biology teacher gives each of the 24 students in the
class a pretest to assess their knowledge of frog anatomy. The teacher wants to compare the effectiveness of an
instructional program in which students physically dissect frogs with the effectiveness of a different program in
which students use computer software that only simulates the dissection of a frog. After completing one of the
two programs, students will be given a posttest to assess their knowledge of frog anatomy. The teacher will then
analyze the changes in the test scores (score on posttest minus score on pretest).
(a) Describe a method for assigning the 24 students to two groups of equal size that allows for a statistically
valid comparison of the two instructional programs.
(b) Suppose the teacher decided to allow the students in the class to select which instructional program on frog
anatomy (physical dissection or computer simulation) they prefer to take, and 11 students choose actual
dissection and 13 students choose computer simulation. How might that self-selection process jeopardize a
statistically valid comparison of the changes in the test scores (score on posttest minus score on pretest) for
the two instructional programs? Provide a specific example to support your answer.
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-7-
2009 AP® STATISTICS FREE-RESPONSE QUESTIONS
4. One of the two fire stations in a certain town responds to calls in the northern half of the town, and the other
fire station responds to calls in the southern half of the town. One of the town council members believes that the
two fire stations have different mean response times. Response time is measured by the difference between the
time an emergency call comes into the fire station and the time the first fire truck arrives at the scene of the fire.
Data were collected to investigate whether the council member’s belief is correct. A random sample of 50 calls
selected from the northern fire station had a mean response time of 4.3 minutes with a standard deviation of
3.7 minutes. A random sample of 50 calls selected from the southern fire station had a mean response time of
5.3 minutes with a standard deviation of 3.2 minutes.
(a) Construct and interpret a 95 percent confidence interval for the difference in mean response times between
the two fire stations.
(b) Does the confidence interval in part (a) support the council member’s belief that the two fire stations have
different mean response times? Explain.
5. For many years, the medically accepted practice of giving aid to a person experiencing a heart attack was to have
the person who placed the emergency call administer chest compression (CC) plus standard mouth-to-mouth
resuscitation (MMR) to the heart attack patient until the emergency response team arrived. However, some
researchers believed that CC alone would be a more effective approach.
In the 1990s a study was conducted in Seattle in which 518 cases were randomly assigned to treatments: 278 to
CC plus standard MMR and 240 to CC alone. A total of 64 patients survived the heart attack: 29 in the group
receiving CC plus standard MMR, and 35 in the group receiving CC alone. A test of significance was conducted
on the following hypotheses.
H 0 : The survival rates for the two treatments are equal.
Ha : The treatment that uses CC alone produces a higher survival rate.
This test resulted in a p-value of 0.0761.
(a) Interpret what this p-value measures in the context of this study.
(b) Based on this p-value and study design, what conclusion should be drawn in the context of this study? Use a
significance level of α = 0.05.
(c) Based on your conclusion in part (b), which type of error, Type I or Type II, could have been made? What is
one potential consequence of this error?
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-8-
2009 AP® STATISTICS FREE-RESPONSE QUESTIONS
STATISTICS
SECTION II
Part B
Question 6
Spend about 25 minutes on this part of the exam.
Percent of Section II score—25
Directions: Show all your work. Indicate clearly the methods you use, because you will be graded on the
correctness of your methods as well as on the accuracy and completeness of your results and explanations.
6. A consumer organization was concerned that an automobile manufacturer was misleading customers by
overstating the average fuel efficiency (measured in miles per gallon, or mpg) of a particular car model. The
model was advertised to get 27 mpg. To investigate, researchers selected a random sample of 10 cars of that
model. Each car was then randomly assigned a different driver. Each car was driven for 5,000 miles, and the
total fuel consumption was used to compute mpg for that car.
(a) Define the parameter of interest and state the null and alternative hypotheses the consumer organization is
interested in testing.
One condition for conducting a one-sample t-test in this situation is that the mpg measurements for the
population of cars of this model should be normally distributed. However, the boxplot and histogram shown
below indicate that the distribution of the 10 sample values is skewed to the right.
sample mean
. What values of that statistic
sample median
(small, large, close to one) might indicate that the population distribution of mpg values is skewed to the
right? Explain.
(b) One possible statistic that measures skewness is the ratio
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-9-
2009 AP® STATISTICS FREE-RESPONSE QUESTIONS
(c) Even though the mpg values in the sample were skewed to the right, it is still possible that the population
distribution of mpg values is normally distributed and that the skewness was due to sampling variability.
To investigate, 100 samples, each of size 10, were taken from a normal distribution with the same mean and
sample mean
was
standard deviation as the original sample. For each of those 100 samples, the statistic
sample median
calculated. A dotplot of the 100 simulated statistics is shown below.
sample mean
was 1.03. Based on the value of 1.03 and
sample median
the dotplot above, is it plausible that the original sample of 10 cars came from a normal population, or do the
simulated results suggest the original population is really skewed to the right? Explain.
In the original sample, the value of the statistic
(d) The table below shows summary statistics for mpg measurements for the original sample of 10 cars.
Minimum
Q1
Median
Q3
Maximum
23
24
25.5
28
32
Choosing only from the summary statistics in the table, define a formula for a different statistic that
measures skewness.
What values of that statistic might indicate that the distribution is skewed to the right? Explain.
STOP
END OF EXAM
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-10-
2009 AP® STATISTICS FREE-RESPONSE QUESTIONS
Probability
Table entry for z is the
probability lying below z.
Table A
z
Standard normal probabilities
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
– 3.4
– 3.3
– 3.2
– 3.1
– 3.0
– 2.9
– 2.8
– 2.7
– 2.6
– 2.5
– 2.4
– 2.3
– 2.2
– 2.1
– 2.0
– 1.9
– 1.8
– 1.7
– 1.6
– 1.5
– 1.4
– 1.3
– 1.2
– 1.1
– 1.0
– 0.9
– 0.8
– 0.7
– 0.6
– 0.5
– 0.4
– 0.3
– 0.2
– 0.1
– 0.0
.0003
.0005
.0007
.0010
.0013
.0019
.0026
.0035
.0047
.0062
.0082
.0107
.0139
.0179
.0228
.0287
.0359
.0446
.0548
.0668
.0808
.0968
.1151
.1357
.1587
.1841
.2119
.2420
.2743
.3085
.3446
.3821
.4207
.4602
.5000
.0003
.0005
.0007
.0009
.0013
.0018
.0025
.0034
.0045
.0060
.0080
.0104
.0136
.0174
.0222
.0281
.0351
.0436
.0537
.0655
.0793
.0951
.1131
.1335
.1562
.1814
.2090
.2389
.2709
.3050
.3409
.3783
.4168
.4562
.4960
.0003
.0005
.0006
.0009
.0013
.0018
.0024
.0033
.0044
.0059
.0078
.0102
.0132
.0170
.0217
.0274
.0344
.0427
.0526
.0643
.0778
.0934
.1112
.1314
.1539
.1788
.2061
.2358
.2676
.3015
.3372
.3745
.4129
.4522
.4920
.0003
.0004
.0006
.0009
.0012
.0017
.0023
.0032
.0043
.0057
.0075
.0099
.0129
.0166
.0212
.0268
.0336
.0418
.0516
.0630
.0764
.0918
.1093
.1292
.1515
.1762
.2033
.2327
.2643
.2981
.3336
.3707
.4090
.4483
.4880
.0003
.0004
.0006
.0008
.0012
.0016
.0023
.0031
.0041
.0055
.0073
.0096
.0125
.0162
.0207
.0262
.0329
.0409
.0505
.0618
.0749
.0901
.1075
.1271
.1492
.1736
.2005
.2296
.2611
.2946
.3300
.3669
.4052
.4443
.4840
.0003
.0004
.0006
.0008
.0011
.0016
.0022
.0030
.0040
.0054
.0071
.0094
.0122
.0158
.0202
.0256
.0322
.0401
.0495
.0606
.0735
.0885
.1056
.1251
.1469
.1711
.1977
.2266
.2578
.2912
.3264
.3632
.4013
.4404
.4801
.0003
.0004
.0006
.0008
.0011
.0015
.0021
.0029
.0039
.0052
.0069
.0091
.0119
.0154
.0197
.0250
.0314
.0392
.0485
.0594
.0721
.0869
.1038
.1230
.1446
.1685
.1949
.2236
.2546
.2877
.3228
.3594
.3974
.4364
.4761
.0003
.0004
.0005
.0008
.0011
.0015
.0021
.0028
.0038
.0051
.0068
.0089
.0116
.0150
.0192
.0244
.0307
.0384
.0475
.0582
.0708
.0853
.1020
.1210
.1423
.1660
.1922
.2206
.2514
.2843
.3192
.3557
.3936
.4325
.4721
.0003
.0004
.0005
.0007
.0010
.0014
.0020
.0027
.0037
.0049
.0066
.0087
.0113
.0146
.0188
.0239
.0301
.0375
.0465
.0571
.0694
.0838
.1003
.1190
.1401
.1635
.1894
.2177
.2483
.2810
.3156
.3520
.3897
.4286
.4681
.0002
.0003
.0005
.0007
.0010
.0014
.0019
.0026
.0036
.0048
.0064
.0084
.0110
.0143
.0183
.0233
.0294
.0367
.0455
.0559
.0681
.0823
.0985
.1170
.1379
.1611
.1867
.2148
.2451
.2776
.3121
.3483
.3859
.4247
.4641
-11-
2009 AP® STATISTICS FREE-RESPONSE QUESTIONS
Probability
Table entry for z is the
probability lying below z.
z
Table A
(Continued)
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
.5000
.5398
.5793
.6179
.6554
.6915
.7257
.7580
.7881
.8159
.8413
.8643
.8849
.9032
.9192
.9332
.9452
.9554
.9641
.9713
.9772
.9821
.9861
.9893
.9918
.9938
.9953
.9965
.9974
.9981
.9987
.9990
.9993
.9995
.9997
.5040
.5438
.5832
.6217
.6591
.6950
.7291
.7611
.7910
.8186
.8438
.8665
.8869
.9049
.9207
.9345
.9463
.9564
.9649
.9719
.9778
.9826
.9864
.9896
.9920
.9940
.9955
.9966
.9975
.9982
.9987
.9991
.9993
.9995
.9997
.5080
.5478
.5871
.6255
.6628
.6985
.7324
.7642
.7939
.8212
.8461
.8686
.8888
.9066
.9222
.9357
.9474
.9573
.9656
.9726
.9783
.9830
.9868
.9898
.9922
.9941
.9956
.9967
.9976
.9982
.9987
.9991
.9994
.9995
.9997
.5120
.5517
.5910
.6293
.6664
.7019
.7357
.7673
.7967
.8238
.8485
.8708
.8907
.9082
.9236
.9370
.9484
.9582
.9664
.9732
.9788
.9834
.9871
.9901
.9925
.9943
.9957
.9968
.9977
.9983
.9988
.9991
.9994
.9996
.9997
.5160
.5557
.5948
.6331
.6700
.7054
.7389
.7704
.7995
.8264
.8508
.8729
.8925
.9099
.9251
.9382
.9495
.9591
.9671
.9738
.9793
.9838
.9875
.9904
.9927
.9945
.9959
.9969
.9977
.9984
.9988
.9992
.9994
.9996
.9997
.5199
.5596
.5987
.6368
.6736
.7088
.7422
.7734
.8023
.8289
.8531
.8749
.8944
.9115
.9265
.9394
.9505
.9599
.9678
.9744
.9798
.9842
.9878
.9906
.9929
.9946
.9960
.9970
.9978
.9984
.9989
.9992
.9994
.9996
.9997
.5239
.5636
.6026
.6406
.6772
.7123
.7454
.7764
.8051
.8315
.8554
.8770
.8962
.9131
.9279
.9406
.9515
.9608
.9686
.9750
.9803
.9846
.9881
.9909
.9931
.9948
.9961
.9971
.9979
.9985
.9989
.9992
.9994
.9996
.9997
.5279
.5675
.6064
.6443
.6808
.7157
.7486
.7794
.8078
.8340
.8577
.8790
.8980
.9147
.9292
.9418
.9525
.9616
.9693
.9756
.9808
.9850
.9884
.9911
.9932
.9949
.9962
.9972
.9979
.9985
.9989
.9992
.9995
.9996
.9997
.5319
.5714
.6103
.6480
.6844
.7190
.7517
.7823
.8106
.8365
.8599
.8810
.8997
.9162
.9306
.9429
.9535
.9625
.9699
.9761
.9812
.9854
.9887
.9913
.9934
.9951
.9963
.9973
.9980
.9986
.9990
.9993
.9995
.9996
.9997
.5359
.5753
.6141
.6517
.6879
.7224
.7549
.7852
.8133
.8389
.8621
.8830
.9015
.9177
.9319
.9441
.9545
.9633
.9706
.9767
.9817
.9857
.9890
.9916
.9936
.9952
.9964
.9974
.9981
.9986
.9990
.9993
.9995
.9997
.9998
-12-
2009 AP® STATISTICS FREE-RESPONSE QUESTIONS
Table entry for p and
C is the point t* with
probability p lying
above it and
probability C lying
between −t * and t*.
Probability p
t*
Table B
t distribution critical values
Tail probability p
df
.25
.20
.15
.10
.05
.025
.02
.01
.005
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
80
100
1000
!
1.000
.816
.765
.741
.727
.718
.711
.706
.703
.700
.697
.695
.694
.692
.691
.690
.689
.688
.688
.687
.686
.686
.685
.685
.684
.684
.684
.683
.683
.683
.681
.679
.679
.678
.677
.675
.674
1.376
1.061
.978
.941
.920
.906
.896
.889
.883
.879
.876
.873
.870
.868
.866
.865
.863
.862
.861
.860
.859
.858
.858
.857
.856
.856
.855
.855
.854
.854
.851
.849
.848
.846
.845
.842
.841
1.963
1.386
1.250
1.190
1.156
1.134
1.119
1.108
1.100
1.093
1.088
1.083
1.079
1.076
1.074
1.071
1.069
1.067
1.066
1.064
1.063
1.061
1.060
1.059
1.058
1.058
1.057
1.056
1.055
1.055
1.050
1.047
1.045
1.043
1.042
1.037
1.036
3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325
1.323
1.321
1.319
1.318
1.316
1.315
1.314
1.313
1.311
1.310
1.303
1.299
1.296
1.292
1.290
1.282
1.282
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
1.711
1.708
1.706
1.703
1.701
1.699
1.697
1.684
1.676
1.671
1.664
1.660
1.646
1.645
12.71
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.080
2.074
2.069
2.064
2.060
2.056
2.052
2.048
2.045
2.042
2.021
2.009
2.000
1.990
1.984
1.962
1.960
15.89
4.849
3.482
2.999
2.757
2.612
2.517
2.449
2.398
2.359
2.328
2.303
2.282
2.264
2.249
2.235
2.224
2.214
2.205
2.197
2.189
2.183
2.177
2.172
2.167
2.162
2.158
2.154
2.150
2.147
2.123
2.109
2.099
2.088
2.081
2.056
2.054
31.82
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
2.718
2.681
2.650
2.624
2.602
2.583
2.567
2.552
2.539
2.528
2.518
2.508
2.500
2.492
2.485
2.479
2.473
2.467
2.462
2.457
2.423
2.403
2.390
2.374
2.364
2.330
2.326
63.66
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845
2.831
2.819
2.807
2.797
2.787
2.779
2.771
2.763
2.756
2.750
2.704
2.678
2.660
2.639
2.626
2.581
2.576
127.3
14.09
7.453
5.598
4.773
4.317
4.029
3.833
3.690
3.581
3.497
3.428
3.372
3.326
3.286
3.252
3.222
3.197
3.174
3.153
3.135
3.119
3.104
3.091
3.078
3.067
3.057
3.047
3.038
3.030
2.971
2.937
2.915
2.887
2.871
2.813
2.807
318.3
22.33
10.21
7.173
5.893
5.208
4.785
4.501
4.297
4.144
4.025
3.930
3.852
3.787
3.733
3.686
3.646
3.611
3.579
3.552
3.527
3.505
3.485
3.467
3.450
3.435
3.421
3.408
3.396
3.385
3.307
3.261
3.232
3.195
3.174
3.098
3.091
636.6
31.60
12.92
8.610
6.869
5.959
5.408
5.041
4.781
4.587
4.437
4.318
4.221
4.140
4.073
4.015
3.965
3.922
3.883
3.850
3.819
3.792
3.768
3.745
3.725
3.707
3.690
3.674
3.659
3.646
3.551
3.496
3.460
3.416
3.390
3.300
3.291
50%
60%
70%
80%
90%
95%
96%
98%
99%
99.5%
99.8%
99.9%
Confidence level C
-13-
.0025
.001
.0005
2009 AP® STATISTICS FREE-RESPONSE QUESTIONS
Probability p
Table entry for p is the point
( χ 2 ) with probability p lying
above it.
(χ2 )
Table C
c 2 critical values
Tail probability p
df
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
80
100
.25
1.32
2.77
4.11
5.39
6.63
7.84
9.04
10.22
11.39
12.55
13.70
14.85
15.98
17.12
18.25
19.37
20.49
21.60
22.72
23.83
24.93
26.04
27.14
28.24
29.34
30.43
31.53
32.62
33.71
34.80
45.62
56.33
66.98
88.13
109.1
.20
1.64
3.22
4.64
5.99
7.29
8.56
9.80
11.03
12.24
13.44
14.63
15.81
16.98
18.15
19.31
20.47
21.61
22.76
23.90
25.04
26.17
27.30
28.43
29.55
30.68
31.79
32.91
34.03
35.14
36.25
47.27
58.16
68.97
90.41
111.7
.15
.10
2.07
3.79
5.32
6.74
8.12
9.45
10.75
12.03
13.29
14.53
15.77
16.99
18.20
19.41
20.60
21.79
22.98
24.16
25.33
26.50
27.66
28.82
29.98
31.13
32.28
33.43
34.57
35.71
36.85
37.99
49.24
60.35
71.34
93.11
114.7
2.71
4.61
6.25
7.78
9.24
10.64
12.02
13.36
14.68
15.99
17.28
18.55
19.81
21.06
22.31
23.54
24.77
25.99
27.20
28.41
29.62
30.81
32.01
33.20
34.38
35.56
36.74
37.92
39.09
40.26
51.81
63.17
74.40
96.58
118.5
.05
3.84
5.99
7.81
9.49
11.07
12.59
14.07
15.51
16.92
18.31
19.68
21.03
22.36
23.68
25.00
26.30
27.59
28.87
30.14
31.41
32.67
33.92
35.17
36.42
37.65
38.89
40.11
41.34
42.56
43.77
55.76
67.50
79.08
101.9
124.3
.025
5.02
7.38
9.35
11.14
12.83
14.45
16.01
17.53
19.02
20.48
21.92
23.34
24.74
26.12
27.49
28.85
30.19
31.53
32.85
34.17
35.48
36.78
38.08
39.36
40.65
41.92
43.19
44.46
45.72
46.98
59.34
71.42
83.30
106.6
129.6
-14-
.02
.01
.005
5.41
7.82
9.84
11.67
13.39
15.03
16.62
18.17
19.68
21.16
22.62
24.05
25.47
26.87
28.26
29.63
31.00
32.35
33.69
35.02
36.34
37.66
38.97
40.27
41.57
42.86
44.14
45.42
46.69
47.96
60.44
72.61
84.58
108.1
131.1
6.63
9.21
11.34
13.28
15.09
16.81
18.48
20.09
21.67
23.21
24.72
26.22
27.69
29.14
30.58
32.00
33.41
34.81
36.19
37.57
38.93
40.29
41.64
42.98
44.31
45.64
46.96
48.28
49.59
50.89
63.69
76.15
88.38
112.3
135.8
7.88
10.60
12.84
14.86
16.75
18.55
20.28
21.95
23.59
25.19
26.76
28.30
29.82
31.32
32.80
34.27
35.72
37.16
38.58
40.00
41.40
42.80
44.18
45.56
46.93
48.29
49.64
50.99
52.34
53.67
66.77
79.49
91.95
116.3
140.2
.0025
9.14
11.98
14.32
16.42
18.39
20.25
22.04
23.77
25.46
27.11
28.73
30.32
31.88
33.43
34.95
36.46
37.95
39.42
40.88
42.34
43.78
45.20
46.62
48.03
49.44
50.83
52.22
53.59
54.97
56.33
69.70
82.66
95.34
120.1
144.3
.001
.0005
10.83
13.82
16.27
18.47
20.51
22.46
24.32
26.12
27.88
29.59
31.26
32.91
34.53
36.12
37.70
39.25
40.79
42.31
43.82
45.31
46.80
48.27
49.73
51.18
52.62
54.05
55.48
56.89
58.30
59.70
73.40
86.66
99.61
124.8
149.4
12.12
15.20
17.73
20.00
22.11
24.10
26.02
27.87
29.67
31.42
33.14
34.82
36.48
38.11
39.72
41.31
42.88
44.43
45.97
47.50
49.01
50.51
52.00
53.48
54.95
56.41
57.86
59.30
60.73
62.16
76.09
89.56
102.7
128.3
153.2