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HW #8 – HT’s and CI’s with t
Note that for each problem parts C through G are:
C) Give the critical value(s) (from the table).
D) Give the value of the test statistic (from the data).
E) Is the answer Yes or No?
F) What is the p-value?
G) Describe the meaning of the p-value in everyday terms.
Also all sample data are given at the end of the assignment.
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. We wish to see if we have good evidence that the mean of all adults is less
than 98.6. Assume the data given at the end of the assignment.
A) Before collecting data, what is the probability of not concluding the mean is less than 98.6
when it actually is less than 98.6?
B) Before collecting data, what is the probability of concluding the mean is less than 98.6 when it
actually is not less than 98.6?
H) Give a 95% CI for the mean body temperature of all adults.
2. (ANSWER GIVEN) Suppose the measurements on the stress needed to break a type of bolt
follow close to a Normal distribution. The advertised mean breaking strength is 80 ksi. We wish
to see if we have good evidence at the 5% significance level that the mean breaking stress of all
bolts is not 80 ksi.
A) Before collecting data, what is the probability of not concluding the mean is not 80 ksi when it
actually is not 80 ksi?
B) Before collecting data, what is the probability of concluding the mean is not 80 ksi when it
actually is 80 ksi?
H) Give a 99% CI for the mean breaking strength of all such bolts.
3. (SOLUTION GIVEN) Assume the cholesterol levels of adult American women can be
described by a Normal model. We wish to see if we have good evidence at the 10% level of
significance that the mean for the population of all adult American women is over 188.
A) Before collecting data, what is the probability of not concluding the mean is over 188 when it
actually is?
B) Before collecting data, what is the probability of concluding the mean is over 188 when it
actually is not?
H) Give a 95% CI for the mean cholesterol level for all adult American women.
4. (HOMEWORK) Biological measurements on the same species often follow a Normal
distribution quite closely. Assume the weights of seeds of a variety of winged bean are
approximately Normal. We wish to see if we have good evidence at the 5% significance level
that the mean is over 525 for this variety.
A) Before collecting data, what is the probability of not concluding the mean is over 525 when it
actually is?
B) Before collecting data, what is the probability of concluding the mean is over 525 when it
actually is not?
H) Give a 99% CI for the mean weights of all seeds of this variety.
5. (ALTERNATE HW) Assume the heights of women aged 20-29 follow approximately a
Normal distribution. We wish to see if we have good evidence at the 1% level of significance
that the mean is not 64 inches.
A) Before collecting data, what is the probability of not concluding the mean is not 64 inches
when it actually is not 64 inches?
B) Before collecting data, what is the probability of concluding the mean is not 64 inches when it
actually is 64 inches?
H) Give a 95% CI for the mean height of all women aged 20-29.
6. (IN CLASS) We wish to see if we have good evidence that the mean time improvement for all
possible participants in a fitness program to run a mile is more than 15 seconds after completing
the program.
A) Before collecting data, what is the probability of concluding the mean is more than 15 seconds
when it actually is not more than 15 seconds?
B) Before collecting data, what is the probability of not concluding the mean is more than 15
seconds when it actually is?
H) Give a 90% CI for the mean improvement for all possible participants.
7. (ANSWER GIVEN) We wish to see if we have good evidence at the 1% significance level that
the mean improvement in the number of sit-ups people could do in 5 minutes before and after an
intense fitness class designed especially abs differs from 20.
A) Before collecting data, what is the probability of concluding the mean is not 20 when it
actually is 20?
B) Before collecting data, what is the probability of not concluding the mean differs from 20
when it actually does?
H) Give a 99% CI for the mean improvement for all possible participants.
8. (SOLUTION GIVEN) We wish to see if there is any difference in the mean weights that two
scales will report for the population of all cans of peaches that could ever be weighed. Use the
5% significance level.
A) Before collecting data, what is the probability of concluding a difference when there is no
difference?
B) Before collecting data, what is the probability of not concluding a difference when there is a
difference?
H) Give a 90% CI for the mean difference for all such cans.
9. (HOMEWORK) We wish to see if we have good evidence at 10% level of significance if there
is any difference in the population of all healthy adult females in the mean absorption into the
blood between a generic drug and the reference name brand drug. We will at random give half
the subjects the generic drug first and the rest will take 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.
A) Before collecting data, what is the probability of concluding a difference when there is no
difference?
B) Before collecting data, what is the probability of not concluding a difference when there is a
difference?
H) Give a 95% CI for the mean difference (Ref – Gen) for all healthy adult females.
10. (ALTERNATE HW) We wish to see at the 5% significance level if there is any difference in
the mean high temperatures in a big city at the airport and downtown.
A) Before collecting data, what is the probability of concluding a difference when there is no
difference?
B) Before collecting data, what is the probability of not concluding a difference when there is a
difference?
H) Give a 95% CI for the mean difference (DT – Airpt) for all possible days.
DATA:
1. SRS, sample size 48, sample mean 98.2, sample standard deviation of .7.
2. SRS, sample size 25, sample mean 75.0, sample standard deviation of 8.3.
3. SRS, sample size 28, sample mean 192.0, sample standard deviation of 24.0..
4. SRS, sample size 22, sample mean 534.0, sample standard deviation of 110.
5. SRS, sample size 21, sample mean 65.2, sample standard deviation of 2.7.
6. SRS
Person 1
2
3
4
5
6
7
8
9
10
After
580 611 542 570 542 540 490 490 488 490
Before 630 660 560 542 580 585 500 522 533 544
Person 11 12 13 14 15 16 17 18 19 20
After
600 465 455 710 600 510 510 480 480 489
Before 520 470 460 700 820 600 610 500 544 566
7. SRS
Person 1
2
3
4
5
6
7
8
9
10 11
After
164 142 154 143 157 147 174 174 163 165 168
Before 150 150 150 94 95 156 160 180 177 99 86
Person 12 13 14 15 16 17 18 19 20 21
After
192 182 195 176 193 184 152 142 190 183
Before 162 162 133 165 165 165 166 166 172 172
Person 22 23 24 25 26 27 28 29 30 31
After
175 146 161 182 183 178 176 192 192 173
Before 153 166 144 153 144 99 80 138 130 111
8. SRS
Can
1
2
3
4
5
6
7
8
9
Scale A 11.83 12.46 11.87 12.99 12.33 13.30 12.73 11.55 13.31
Scale B 11.71 12.44 11.91 12.58 11.88 13.49 13.11 11.02 12.99
Can
12
13
14
15
16
17
18
19
20
Scale A 12.41 12.51 12.14 12.17 12.80 12.27 11.57 12.57 11.59
Scale B 12.78 12.38 11.68 11.95 12.81 12.38 11.36 11.48 11.50
9. SRS
Subject
A
B
C
D
E
F
G
H
I
Reference 4110 2536 2769 3853 1832 2436 1999 1719 1829
Generic
1755 1148 1603 2254 1309 2120 1851 1878 1685
Subject
K
L
M
N
O
P
Q
R
S
Reference 2354 1864 1022 2256 938
1339 1262 1438 1735
Generic
2738 2202 1254 3051 1287 1930 1964 2549 3335
10. SRS
Day
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Downtown 72 74 61 90 88 46 52 60 70 44 32 60 60 45
Airport
75 73 61 94 93 45 52 60 68 51 35 58 59 49
10
11
12.26 12.13
11.58 12.07
21
11.64
11.45
J
2594
2643
T
920
3044
15 16 17
93 97 80
93 96 84