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# Probabilities, Confidence Intervals and Binomial Random Variables

8.50 Refer to Exercise 8.25, vhere we reported on the results of Harris polls conducted in 1983 and 1992. The 1992 results summarized the responses of n ? 1251 individuals, whereas we assumed that the results for the 1983 poll involved n = 1250 respondents. The data are summarized in the accompanying table for your convenience.
Health Issue 1983 Survey 1992 Survey
Ate recommended amount of fibrous food .59 .53
Avoided fat .55 .51
Avoided excess salt .53 .46
Used scatbelts .19 .70
Used smoke detectors .67 .90
a Construct a 98% confidence interval for the difference in the proportions of Americans who ate the recommended amounts of fibrous foods in 1983 and in 1992.
b Based upon the interval you obtained in (a), do you think that the proportion of Americans who ate (he recommended amount of fibrous foods decreased between 1983 and 1992? Explain.
8.54 Alice Chang and her colleagues (A. Chang, T. L. Rosenthal, R. H. Bryant, R. Fl. Rosenthal. R. M. Heidlage, and B. K. Fritzler. 'Companng High School and College Students' Leisure in- Wrests and Stress Ratings," Behavioral Research Therapy 31(2), 1993) tested 559 high school students using a leisure-interest checklist (LIC) survey. The means and standard deviations for each of the seven LIC factor scales are given in the following table for n = 252 male students and n = 307 female students.
a Give a 95% confidence interval for the difference in the means for male and female students for the cultural activity scale. Interpret this interval.
8.58 Let Y be a binomial random variable with parameter p. Find the sample size necessary to 1
estimate p to within .05 with probability .95 in the following situations:
a if p is thought to be approximately .9.
b if no information about p is known (usc p = .5 in estimating the variance of ).

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