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# confidence interval and hypothesis test

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Heights of Winners and Runners-Up Listed below are the heights of candidates who won presidential elections and the heights of the candidates with the next highest number of popular votes. The data are in chronological order, so the corresponding heights from the two lists are matched. For candidates who won more than once, only the heights from the first election are included, and no elections before 1990 are included.
a. A well-known theory is that winning candidates tend to be taller than the corresponding losing candidates. Use a 0.05 significance level to test that theory. Does height appear to be an important factor in winning the presidency?
b. If you plan to test the claim in part (a) by using a confidence interval, what confidence level should be used? Construct a confidence interval using that confidence level, then interpret the result.
Won Presidency Runner-Up
----------------------------------------------- ---------------------------------------------
71 74.5 74 73 69.5 71.5 75 72 73 74 68 69.5 72 71 72 71.5
70.5 69 74 70 71 72 70 67 70 68 71 72 70 72 72 72

Measuring Intelligence in Children Mental measurements of young children are often made by given them blocks and telling them to build a tower as tall as possible, One experiment of block building was repeated a month later, with the times (in seconds) in listed in the accompanying table (based on data from "Tower Building," by Johnson and Courtney, Child Development, Vol. 3).
a. Is there sufficient evidence to support the claim that there is a difference between the two times? Use a 0.01 significance level.
b. Construct a 90% confidence interval for the mean of the differences. Do the confidence interval limits contain 0, indicating that there is not a significant difference between the times of the first and the second trials?

Child A B C D E F G H I J K L M N O
First trial 30 19 19 23 29 178 42 20 12 39 14 81 17 31 52
Second trial 30 6 14 8 14 52 14 22 17 8 11 30 14 17 15

Bipolar Depression Treatment In clinical experiments involving different groups of independent samples, it is important that the groups be similar in the important ways that affect the experiment. In an experiment designed to test the effectiveness of paroxetine for treating bipolar depression, subjects were measured using the Hamilton depression scale with the results given below (based on data from "Double-Blind Placebo-Controlled Comparison of Impipramine and Paroxetine in the Treatment of Bipolar Depression, " by Nemeroff et al., American Journal of Psychiatry, Vol. 158, No. 6). Using a 0.05 significance level, test the claim that both populations have th same standard deviation. Based on the results, does it appear that the two populations have different standard deviations?
Placebo group: n = 43, = 21.57, s = 3.87
Paroxetine treatment group: n = 33, = 20.38, s = 3.91

Effect of Birth Weight on IQ Score When investigating a relationship between birth weight and IQ, researchers found 258 subjects with extremely low birth weights (less than 1000 g) had Wechsler IQ score at age 8 with a mean of 95.5 and a variance of 256.0. For 220 subjects with normal birth weights, the mean at age 8 is 104.9 and the variance is 198.8 (based on data from "Neurobehavioral Outcomes of School-age Children Born Extremely Low Birth Weight or Very Preterm in the 1990s," by Anderson et al., Journal of American Medical Association, Vol. 289, No. 24). Using a 0.05 significance level, test the claim that babies with extremely low birth weights and babies with normal birth weights have different amounts of variation. (Hint: The conclusion is not clear from Table A-5, so use this upper value: F = 1.2928)