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 / 997651 / 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 / Right thread / Left thread / Subject / Right thread / Left thread1 / 113 / 137 / 14 / 107 / 87
2 / 105 / 105 / 15 / 118 / 166
3 / 130 / 133 / 16 / 103 / 146
4 / 101 / 108 / 17 / 111 / 123
5 / 138 / 115 / 18 / 104 / 135
6 / 118 / 170 / 19 / 111 / 112
7 / 87 / 103 / 20 / 89 / 93
8 / 116 / 145 / 21 / 78 / 76
9 / 75 / 78 / 22 / 100 / 116
10 / 96 / 107 / 23 / 89 / 78
11 / 122 / 84 / 24 / 85 / 101
12 / 103 / 148 / 25 / 88 / 123
13 / 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.