AP STATS REVIEW – PROBABILITY
Multiple Choice
Chapter 6
1. D
2. B
3. B
4. C
5. C
6. D
7. D
8. C
9. D
10. C
Chapter 7
1. B
2. C
3. B
4. E
5. A
6. E
7. B
8. A
9. D
10. C
Chapter 8
1. D
2. E
3. A
4. C
5. B
6. C
7. C
8. B
9. E
10. A
11. B
Chapter 9
1. C
2. C
3. B
4. B
5. E
6. C
7. C
8. C
9. B
(2002B #2)
Part (a) is
Essentially correct if computes P(X ≤ 38) (except for minor arithmetic errors)
Partially correct if computes P(X = 38) = 0.16 or P(X > 38) = .08 or P(X < 38) = 0.76 or P(X ≥ 38) = 0.24 or
gives 0.92 but does not show any work.
Part (b) is
Essentially correct if (except for minor arithmetic errors) correctly computes expected value for number of no
shows and indicates fully where the 36.9 and 4.1 come from.
Rounding this value to 4 is considered a minus, though can be forgiven with “about” or “approximately.”
Partially correct if correctly computes expected value of X = number of passengers who show up for the flight
(instead of no shows) and shows work,
OR if incorrectly computes the first expected value but “subtracts,”
OR if correctly computes the first expected value but “subtracts” from the wrong number,
OR if does not show work for 36.9 but subtracts to get 4.1.
Part (c) is
Essentially correct if correctly computes the conditional probability. Complete notation could be considered a
“plus.”
Partially correct if correctly computes P(X = 36 | X < 41) = .46/.49 = 0.465 or P (X=36 | X≤38) = .46/.92 = 0.5
or incorrectly tries to solve the conditional probability (e.g., multiplies probabilities in numerator).
(2003B #2)
Part (a) is scored as either essentially correct (E) (may be minor arithmetic errors) or incorrect (I).
Part (b) is
Essentially correct (E) if the conditional probability is correctly calculated.
Partially correct (P) if the student reverses the conditioning, calculating
P (income over 50000 | age 31-45) = 35/89 = 0.3933
OR
calculates the correct probability for the wrong column, e.g., 32/64
Part (c) is
Essentially correct (E) if the student
1. indicates that the two variables are not independent
AND
2. the explanation is tied to the fact that the probabilities in parts (a) and (b) are not equal (the answer must
be based on parts (a) and (b))
Partially correct (P) if the student indicates that the two variables are not independent, but the explanation is
incorrect, or is not based on the answers to parts (a) and (b); i.e., performing new correct calculations instead of
referring to those in parts (a) and (b).
(2012 #2)
Part (a):
1 for correct answer (including Binomial calculation) = 0.4219
0.5 if answer is ¾ or (1/4)^3 = 0.0156 or 0.4219 with no work
Part (b):
1 if the expected value, 800, is correct (except for minor computational errors)
0.5 if expected value is computed as
 800 + expected winnings on one spin, or 800 + 200 = 1000
 E(outcome on one spin)=200 but then solution breaks down
 E(winnings on 4th spin)=0 but then solution breaks down
 With fairly major computational errors e.g., 3200/3
Part (c):
0.5 for each element of the test that is correct
1.
2.
3.
4.
statement of hypotheses
identification of test and check of sample size condition
correct mechanics : chi-squared = 4.24, df = 3, p-value = 0.2367
statement of conclusion (fail to reject)
If both an α and a p-value are given, the linkage is implied. If no α is given, the solution must be explicit about
the linkage by giving a correct interpretation of the p-value or explaining how the conclusion follows from the pvalue.
NOTE: If the p-value in element 3 is incorrect but the conclusion is consistent with the computed p-value,
element 4 can be considered as correct.
(2006 #3)
Part (a) is essentially correct (E) if the student clearly does ALL three of the following:
 identifies the distribution as normal;
 specifies BOTH m and s; AND
 calculates the correct probability.
Part (a) is partially correct (P) if the student:
 calculates the correct probability but fails to identify the distribution as normal with BOTH m and s
specified; OR
 correctly identifies the distribution as normal with BOTH m and s specified but fails to calculate the
correct probability.
Notes:


The student may use the distribution of the error, E, to solve the problem. That is, finding the area
below -2 for a normal distribution with mean 0 and standard deviation 1.5 should be scored essentially
correct
If only the calculator command normalcdf (-9999, 0, 2, 1.5) is provided along with the probability
0.0912, then the response should be scored as partially correct (P).
Part (b) is essentially correct (E) if the student calculates the correct probability AND:
 correctly applies complement and probability rules using the value obtained in part (a); OR
 clearly identifies the distribution as binomial AND specifies BOTH n and p using the value obtained in
part (a).
Part (b) is partially correct (P) if the student:
 clearly identifies the distribution as binomial AND specifies BOTH n and p, using the value obtained in
part (a), but does not calculate the correct probability; OR
 clearly identifies the distribution as binomial AND specifies BOTH n and p using a value of p that is
unrelated to the value obtained in part (a) and calculates the correct probability based on their value of
p; OR
 calculates the correct probability using the value obtained in part (a) but fails to correctly identify the
distribution as binomial with BOTH n and p specified; OR
 recognizes the solution as the sum of the product of the probabilities of successes and failures, using the
answer from part (a), but omits only the binomial coefficients.
Notes:
 The solution using the binomial distribution with p = 0.0918
 If only the calculator command 1 - binomcdf (3, 0.0918, 0) is provided along with the probability
0.2509, then the response should be scored as partially correct (P).
Part (c) is essentially correct (E) if the student clearly does ALL three of the following:
 identifies the distribution of the sample mean as normal;
 specifies BOTH mx and sx ; AND
 calculates the correct probability.
Part (c) is partially correct (P) if the student:
 calculates the correct probability, but fails to identify the distribution of the sample mean as normal
withBOTH mx and sx specified; OR
 correctly identifies the distribution of the sample mean as normal with BOTH mx and sx specified, but
fails to calculate the correct probability.
(2006B #3)
Part (a) is essentially correct (E) if the student clearly shows ALL three of the following:
 Indicates the distribution is normal.
 Specifies BOTH the mean, m, and the standard deviation, s .
 Calculates the correct probability.
Part (a) is partially correct (P) if the student:
 Calculates the correct probability but fails to specify BOTH m and σ and in the identification of the
normal the distribution OR
 Completely identifies the distribution as normal with BOTH m and σ correctly specified but fails to
calculate the correct probability, e.g., calculates the probability that the ball travels less than 291.2 yards
OR
 Completely identifies the distribution as normal with BOTH m and σ correctly specified but uses the
empirical rule to provide an approximate answer.
Notes:
 Calculator solution is 0.1265. If this is the only information provided, the response is scored as incorrect
(I).
 If only the calculator command Normalcdf (-9999, 291.2, 288, 2.8) is provided along with 0.1265, then
the response should be scored as partially correct (P).
 Minor arithmetic or transcription errors will not necessarily lower the score.
Part (b) is essentially correct (E) if the student calculates the correct probability and:
 Clearly indicates the distribution is binomial AND specifies both n and p using the value obtained in
part (a) OR
 Correctly applies complement and probability rules using the value obtained in part (a).
Part (b) is partially correct (P) if the student:



Clearly indicates the distribution is binomial AND specifies both n and p using the value obtained in
part (a), but does not calculate the probability correctly, OR
Calculates the correct probability using the value obtained in part (a) but fails to completely identify the
distribution as binomial with both n and p specified, OR
Indicates a correct procedure for computing the probability but uses a value of p that is different from
the value obtained in part (a).
Notes:
If only the calculator command is provided along with the probability 0.4915, then the response should be scored
as partially correct.
Part (c) is essentially correct (E) if the student clearly shows ALL three of the following:
 Identifies the 99th percentile for the standard normal distribution
 Sets up an appropriate equation
 Solves for the desired mean
Part (c) is partially correct (P) if the student:
 Recognizes that the 99th percentile for the standard normal distribution must be used and sets up the
appropriate equation but does not solve the equation for the desired mean. OR
 Sets up an appropriate equation using the correct minimum distance but provides an incorrect mean
because an incorrect upper tail percentile (say the 99.5th percentile) was used, OR
 Sets up an appropriate equation using the correct percentile of the standard normal distribution but
provides an incorrect mean because an incorrect minimum distance was used (say 288 yards) was used.
]
(2007B #2)
This question is scored in four sections. Each section is scored as either essentially correct (E), partially correct
(P), or incorrect (I).
Section 1 is part (a), section 2 is part (b), and sections 3 and 4 consist of elements of part (c). This scoring gives
part (c) double weight relative to either part (a) or part (b).
Section 1 is essentially correct (E) if P( X >3) is correctly computed and work is shown in part (a).
Section 1 is partially correct (P) if:
 P( X ≥3) = 0.26 is computed; OR
 a correct numerical answer is given but no work is shown.
Section 2 is essentially correct (E) if in part (b):
 the probability from part (a) is correctly used to calculate the probability that exactly 2 households are
in violation, either using the binomial pdf or using general probability rules, AND
 work is shown.
Section 2 is partially correct (P) if in part (b):
 the student computes P(Y ≥ 2) = 0.5270 or P(Y ≤ 2) = 0.7659 instead of P(Y = 2); OR
 the correct probability is given but no work is shown; OR
 the binomial coefficient is omitted (0.83)8 (0.17)2 = 0.0065.
Section 3 is essentially correct (E) if the response to part (c):
 recognizes that the distribution of X will be approximately normal; OR
 the response says that the distribution of X is more symmetric than the population distribution AND
mentions that the population distribution is highly skewed.
Section 3 is partially correct (P) if the response to part (c) reports a normal distribution for X without indicating
that the normal distribution is an approximation.
Section 4 is essentially correct (E) if the response to part (c) provides the appropriate mean 1.65 and
standard deviation 0.1511.
Section 4 is partially correct (P) if the response to part (c):
 provides either the correct mean or the correct standard deviation for X , but not both; OR
 provides correct numerical values for both the mean and standard deviation but sample notation ( X and
s) is used instead of population notation (μ and / X σ =σ n ); OR
 says only that the X distribution is centered in the same place as the population and has a smaller
standard deviation than the population (and does not give the values of 1.65 and 0.1511).
(2008B #5)
Part (a) is divided into two sections: section 1 and section 2. Section 1 is scored as essentially correct (E) or
incorrect (I). Section 2 is scored as essentially correct (E), partially correct (P), or incorrect (I).
Section 1 is scored as follows:
Essentially correct (E) if the response states that the distribution of Y – X is normal.
Section 2 is scored as follows:
Essentially correct (E) if the response correctly computes the mean and standard deviation AND shows some
work for the calculation of the standard deviation. May contain a minor arithmetic error.
Partially correct (P) if the response correctly states the values of the mean and standard deviation.
Part (b) constitutes section 3 and is scored as essentially correct (E), partially correct (P), or incorrect (I).
Section 3 is scored as follows:
Essentially correct (E) if the response uses the distribution information from part (a) to correctly compute the
desired probability. If the mean or standard deviation is computed incorrectly in part (a), those values should be
used in part (b). (Note: If variances are incorrectly subtracted instead of added, σ = 17.32, z = −1.73 , and the
probability is 1− 0.0418 = 0.9582 .)
Partially correct (P) if the response computes P(Y − X < 0) instead of P(Y − X > 0) and gets P(Y − X < 0) = 1.0 −
0.9099 = 0.0901.
Part (c) constitutes section 4 and it is scored as essentially correct (E), partially correct (P), or incorrect (I).
Section 4 is scored as follows:
Essentially correct (E) if the response correctly concludes that the train from Bullsnake should be delayed by
about 82 minutes. If the mean or standard deviation is computed incorrectly in part (a), those values should be
used in part (c). (Note: If variances are subtracted instead of added, the delay time will be (2.33)(17.32) + 30 =
70.36.)
Partially correct (P) if a correct line of reasoning is explored but the student fails to reach the correct answer.
(2011 #2)
Scoring
Parts (a), (b) and (c) are scored as essentially correct (E), partially correct (P) or incorrect (I).
Part (a) is scored as follows:
Essentially correct (E) if the response has the correct conditional probability AND
shows the work.
Partially correct (P) if the response has the correct reverse conditional probability (of
being a male given that he is registered for Party Y),
OR
if the response has the correct conditional probability BUT does not show work.
Incorrect (I) if the response fails to meet the criteria for E or P.
Part (b) is scored as follows:
Essentially correct (E) if the response identifies two values whose inequality implies a
lack of independence between the events AND includes the following three
components:
1. Correct computations of the two values.
2. An explicit statement of whether the two values are equal or unequal.
3. An appropriate conclusion about the independence of the events.
Partially correct (P) if the response identifies two values whose inequality implies a
lack of independence between the events but includes only two of the three
components listed above.
Incorrect (I) if the response fails to meet the criteria for E or P.
Part (c) is scored as follows:
Essentially correct (E) if the response shows the same conditional distribution of party
registration for both males and females AND includes the following two components:
1. Correct proportions for each party.
2. Correct labels (Party W, Party X, Party Y).
Partially correct (P) if the response shows the same conditional distribution of party
registration for both males and females AND includes only one of the two
components listed above.
Incorrect (I) if the response fails to meet the criteria for E or P.
Note: For all three parts, an incorrect statement that indicate a serious
misunderstanding of statistical concepts, even if unrelated to the rest of the response,
lowers the score one level (that is, from E to P, or from P to I). An example of this is a
response that indicates confusion between independent events and disjoint events.
(2013P #5)
Scoring Parts (a), (b), and (c) are scored as essentially correct (E), partially correct (P), or incorrect
(I).
Part (a) is scored as follows:
Essentially correct (E) if student (1) says the sampling distribution will be approximately
normal, (2) gives a correct value for the mean, and (3) gives a correct value for the standard
deviation.
Partially correct (P) if student answers two of the three parts (sampling distribution shape,
mean, and standard deviation) correctly.
Incorrect (I) if the student answers only one part correctly OR The student is clearly talking
about the data distribution, the sample distribution, or the population distribution.
Note: If the student says that the sampling distribution is normal rather than approximately
normal, this can be overlooked.
Part (b) is scored as follows:
Essentially correct (E) if the probability is calculated correctly with a reasonable sketch or
evidence of calculation shown.
Partially correct (P) if the wrong standard deviation is used to compute the probability and a
reasonable sketch or evidence of calculation is shown.