8.39 Can We Use the Large-Sample Confidence Interval?

Chapter 8 Section 2

Homework A

8.39 Can we use the large-sample confidence interval?

In each of the following circumstances state whether you should use the large-sample confidence interval.

(a) n1 =30, n2 = 30, X1 = 10, and X2 = 15.

(b) n1 =15, n2 = 10, X1 = 10, and X2 = 5.

(d) n1 = 40, n2= 40, X1 = 20, and X2= 12.

(e) n1= 50, n2= 50, X1= 40, and X2. = 45.

8.41 Comparing cell phone ownership in 2003 and 2004. In Exercise 8.4 (page 497), you were asked to compare the 2004 proportion of cell phone owners (89%) with the 2003 estimate (83%). It would be more appropriate to compare these two proportions using the methods of this section. Given that the sample size of each SRS is 1200 students, compare these two years with a significance test, and give an estimate of the difference in proportions of undergraduate cell phone owners with a 95% margin of error. Write a short summary of your results. Show all your work that leads to the calculation of the p-value, state the null and alternative hypothesis.

8.42 - and the Normal distribution. Supposethere are two binomial populations. For the first, thetrue proportion of successes is 0.3; for the second,it is 0.4. Consider taking independent samples fromthese populations, 50 from the first and 60 from the second.

(a) Find the mean and the standard deviation of the distribution of - .

(b) This distribution is approximately Normal. Sketch this Normal distribution and mark the location of the mean.

(c) Find a value d for which the probability is 0.95 that the difference in sample proportions is within ±d. Mark these values on your sketch.

8.46 A comparison of the proportion of frequent binge drinkers. In the published report on binge drinking that we used for Example 8.1, survey results from both 1993 and 1999 are presented. Using the table below, test whether the proportions of frequent binge drinkers are different at the 5% level. Also construct a 95% confidence interval for the difference. Write a short summary of your results.

Year / N / X
1993 / 14,995 / 2973
1999 / 13,819 / 3140

8.48 Effects of reducing air pollution. A study that evaluated the effects of a reduction in exposure to traffic-related air pollutants compared respiratory symptoms of 283 residents of an area with congested streets with 165 residents in a similar area where the congestion was removed because a bypass was constructed. The symptoms of the residents of both areas were evaluated at baseline and again a year after the bypass was completed.24 For the residents of the congested streets, 17 reported that their symptoms of wheezing improved between baseline and one year later, while 35 of the residents of the bypass streets reported improvement.

(a) Find the two sample proportions.

(b) Report the difference in the proportions and the standard error of the difference.

(c) What are the appropriate null and alternative hypotheses for examining the question of interest? Be sure to explain your choice of the alternative hypothesis.

(d) Find the test statistic. Construct a sketch of the distribution of the test statistic under the assumption that the null hypothesis is true. Find the P-value and use your sketch to explain its meaning.

(e) Is no evidence of an effect the same as evidence that there is no effect? Use a 95% confidence interval to answer this question. Summarize your ideas in a way that could be understood by someone who has very little experience with statistics.

(f) The study was done in the United Kingdom. To what extent do you think that the results can be generalized to other circumstances?

8.53 Gender bias in textbooks. To what extent do syntax textbooks, which analyze the structure of sentences, illustrate gender bias? A study of this question sampled sentences from 10 texts.27 One part of the study examined the use of the words "girl," "boy," "man," and "woman." We will call the first two words juvenile and the last two adult. Is the proportion of female references that are juvenile (girl) equal to the proportion of male references that are juvenile (boy)? Here are data from one of the texts:

48 52

(a) Find the proportion of juvenile references for females and its standard error. Do the same for the males.

(b) Use a test of significance to examine whether the two proportions are equal.