12.1 Tests about a Population Mean (pp.742-765)
1. State the null hypothesis for a one sample t-test.
2. State and use diagrams to illustrate the three possible alternative hypotheses for a one sample t-test
3. Give the formula for the one-sample t-statistic, and define each variable in the equation.
4. How is the t-statistic interpreted?
5. What information would lead us to apply a paired t-test to a study, and what would be the statistic
of interest?
6. What critical value 𝑡 ∗ from the table C satisfies each of the following conditions?
(a) The t-distribution with 5 degrees of freedom has probability 0.5 to the right of 𝑡 ∗ .
(b) The t-distribution with 21 degrees of freedom has probability 0.99 to the left of 𝑡 ∗.
7. The on sample t statistic for testing 𝐻0 : 𝜇 = 0 𝑎𝑛𝑑 𝐻𝑎 : 𝜇 > 0 from a sample of n = 15 observations
has the value t = 1.82.
(a) What are the degrees of freedom for this statistic?
(b) Give the two critical value 𝑡 ∗ from table C that bracket t. What are the upper tail probabilities p for
these two entries?
(c) Between what two values does the P-value of the test fall?
(d) Is the value t = 1.82 significant at the 5% level? Is it significant at the 1% level?
8. Here are estimates of the daily intakes of calcium (in milligrams) for 38 women between the ages of
18 and 24 years who participated in a study of women’s bone health:
808
882
1062
970
909
802
374
416
784
997
651
716
438
1420
1425
948
1050
976
572
403
626
774
1253
549
1325
446
465
1269
671
696
1156
684
1933
748
1203
2433
1255
1100
Suppose that the recommended daily allowance (RDA) of calcium for women of this age range is 1200
milligrams. Doctors involved in the study suspect that participating subjects had significantly lower
calcium intakes that the RDA.
(a) Test the doctors claim at the 𝛼 = 0.05 significance level. Follow the Inference Toolbox.
(b) Remove any outliers from the data and run the test in (a) again. Explain any differences in your
results.
9. The design of controls and instruments affects how easily people can use them. A student project
investigated this effect by asking 25 right-handed students to turn a knob (with their right hands) that
moved an indicator by screw action. There were two identical instruments, one with a right-handed
thread (the knob turns clockwise) and the other with a left-hand thread (the knob turns
counterclockwise). The following table gives the times in seconds each subject took to move the
indicator a fixed distance:
Subject
1
2
3
4
5
6
7
8
9
Right thread
113
105
130
101
138
118
87
116
75
Left thread
137
105
133
108
115
170
103
145
78
Subject
14
15
16
17
18
19
20
21
22
Right thread
107
118
103
111
104
111
89
78
100
Left thread
87
166
146
123
135
112
93
76
116
10
11
12
13
96
107
23
89
78
122
84
24
85
101
103
148
25
88
123
116
147
(a) Each of the 25 students used both instruments. Discuss briefly how you would use
randomization in arranging the experiment.
(b) The project designers hoped to show that right handed people find right-handed threads easier
to use. Carry out a significance test at the 5% level to investigate this claim.
(c) Describe a Type I and Type II error in this situation, and give possible consequences of each.
(d) How likely is the significance test in part (b) to detect a mean difference of 5 seconds between
the left-hand and right-hand times? Is this sufficient power?
(e) Construct and interpret a 90% confidence interval for the mean time advantage of right-handed
over left-handed threads in this situation.
(f) Do you think that the time saved would be of practical importance if the task were performed
many times – form example, by an assembly line worker? To help answer this question, find the
mean time for right-handed threads as a percent of the mean time for left-handed threads.