HW #18 – Introduction to Confidence Intervals
1. (IN CLASS) Most people think that the average body temperature in adult humans is 98.6. However, this
figure is based on data from the 1800’s. In a 1992 article in the Journal of the American Medical Association, it
is reported a more accurate figure is 98.2. Assume a normal model is appropriate and that the standard deviation
is 0.7. Assume the standard deviation is from the population.
A) Assume the 98.2 was obtained from a sample of size 258. Give a 95% confidence interval for the mean body
temperature for all adult humans.
B) How large a sample is needed to ensure that a 95% confidence interval will have a margin of error of only 0.05
degrees?
2. (ANSWER GIVEN) Suppose the measurements on the stress needed to break a type of bolt follow a Normal
distribution with a mean of 75 kilopounds per square inch(ksi) and a standard deviation of 8.3 ksi. Assume the
standard deviation is from the population.
A) Assume the estimate of the mean of 75 came from a sample of size 410. Give a 90% confidence interval for the
mean of all such bolts.
B) How large a sample is needed to ensure that a 90% confidence interval will have a margin of error of only 0.5 ksi?
3. (SOLUTION GIVEN) Assume the cholesterol levels of adult American women can be described by a Normal model
with a mean of 188 mg/dL and standard deviation of 24. Assume the standard deviation is from the population.
A) Assume this mean of 188 came from a sample of size 508. Give a 99% confidence interval for the mean cholesterol
level of all adult American women.
B) How large a sample is needed to ensure that a 99% confidence interval will have a margin of error of only 1 mg/dL?
4. (HOMEWORK) Biological measurements on the same species often follow a Normal distribution quite closely. The
weights of seeds of a variety of winged bean are approximately Normal with a mean of 525 mg and a standard
deviation of 110 mg. Assume the standard deviation is from the population.
A) Assume that this mean of 525 came from a sample of size 120. Give a 95% confidence interval for the mean
weights of all seeds.
B) How large a sample is needed to ensure that a 95% confidence interval will have a margin of error of 10 mg?
5. (ALTERNATE HW) The heights of women aged 20-29 follow approximately a Normal distribution with a mean of
64 inches and a standard deviation of 2.7 inches. Assume the standard deviation is from the population.
A) Assume the mean of 64 inches came from a sample of size 911. Give a 95% confidence interval for the mean height
of all such women.
B) How large a sample is needed to ensure that a 95% confidence interval will have margin of error of 0.5 inches?
6. (IN CLASS) Given are the differences in the times a person could run 1 mile before and after an intense
fitness class. Times are in seconds.
Person
After
Before
Person
After
Before
1
580
630
11
600
520
2
611
660
12
465
470
3
542
560
13
455
460
4
570
542
14
710
700
5
542
580
15
600
820
6
540
585
16
510
600
7
490
500
17
510
610
8
490
522
18
480
500
9
488
533
19
480
544
10
490
544
20
489
566
Assume the standard deviation of the improvements in the population is 70 seconds.
A) Give a 95% CI for the mean improvement in times (after – before) for all possible people.
B) How large a sample is needed so that the margin of error of a 95% CI will be 15 seconds?
7. (ANSWER GIVEN) The weights of 21 randomly selected cans of peaches are weighed on two scales.
Can
1
2
3
4
5
6
7
8
9
10
11
Scale A 11.83 12.46 11.87 12.99 12.33 13.30 12.73 11.55 13.31 12.26 12.13
Scale B 11.71 12.44 11.91 12.58 11.88 13.49 13.11 11.02 12.99 11.58 12.07
Can
12
13
14
15
16
17
18
19
20
21
Scale A 12.41 12.51 12.14 12.17 12.80 12.27 11.57 12.57 11.59 11.64
Scale B 12.78 12.38 11.68 11.95 12.81 12.38 11.36 11.48 11.50 11.45
Assume the standard deviation of the differences in the population is .15.
A) Give a 95% CI for the mean A-B for the population.
B) How large a sample is needed so that the margin of error of a 95% CI is .05?
8. (SOLUTION GIVEN) Given are the number of sit-ups people could do in 5 minutes before and after an
intense fitness class designed especially abs.
Person
After
Before
Person
After
Before
Person
After
Before
1
164
150
12
192
162
22
175
153
2
142
150
13
182
162
23
146
166
3
154
150
14
195
133
24
161
144
4
143
94
15
176
165
25
182
153
5
157
95
16
193
165
26
183
144
6
147
156
17
184
165
27
178
99
7
174
160
18
152
166
28
176
80
8
174
180
19
142
166
29
192
138
9
163
177
20
190
172
30
192
130
10
165
99
21
183
172
31
173
111
11
168
86
Assume the standard deviation of after – before in the population is 40.
A) Give a 95% CI for the mean after – before for the population.
B) How large a sample is needed so that the margin of error of a 95% CI is 3?
9. (HOMEWORK) The table gives data on the absorption into the blood taken on 20 healthy female
subjects for a pair of drugs, one generic and the other the reference name brand drug. Half were picked at
random and received the generic drug first and the rest took the reference drug first. In all cases, a washout
period separated the two drugs so that the first had disappeared before the subject took the second.
Subject
Reference
Generic
Subject
Reference
Generic
A
4110
1755
K
2354
2738
B
2536
1148
L
1864
2202
C
2769
1603
M
1022
1254
D
3853
2254
N
2256
3051
E
1832
1309
O
938
1287
F
2436
2120
P
1339
1930
G
1999
1851
Q
1262
1964
H
1719
1878
R
1438
2549
I
1829
1685
S
1735
3335
J
2594
2643
T
920
3044
Assume the standard deviation of the differences in the population is 1000.
A) Give a 95% CI for the mean difference (Ref – Gen) for all people.
B) How large a sample is needed to ensure the margin of error of a 95% CI for the difference is 50?
The table gives the high temperature on a random sample of 17 days at the downtown
and airport locations in a big city. Assume a normal population and a standard deviation of the differences
in the population of .35 degrees.
Day
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17
Downtown 72 74 61 90 88 46 52 60 70 44 32 60 60 45 93 97 80
Airport
75 73 61 94 93 45 52 60 68 51 35 58 59 49 93 96 84
10.(ALTERNATE HW)
A) Give a 95% CI for the mean difference (downtown – airport) for all days.
B) How large a sample is needed to ensure the margin of error of a 95% CI for the difference is .05 degrees?
.