Question

Recall from Activities 16-1 and 16-3 that the Kaiser Family Foundation commissioned an extensive survey in 2004 that investigated the degree to which American youths aged 8–18 have access to various forms of media. Of the 1036 girls in the sample, 64% had a television in their bedrooms, compared to 72% of the 996 boys in the sample.

a. Suppose you want to use these sample results to produce a 95% confidence interval for πg - πb. Describe in words what πg - πb represents.

b. Calculate this interval, and interpret what it reveals. Be sure to mention whether the interval contains all negative values, all positive values, or some of each.

c. Calculate a 99% confidence interval for πg - πb. Comment on how its midpoint and width compare to the 95% interval

note: TT= pi

Answer 1

a) The interpretation of the confidence interval πg - πb is that the difference between the two population proportions lies in the given confidence interval with 0.95 probability.

b) The pooled proportion here is computed as:
P = (0.64*1036 + 0.72*996) / (1036 + 996) = 0.6792

The standard error now is computed here as:

Now from standard normal tables, we have:
P(-1.96 < Z < 1.96) = 0.95

Therefore the confidence interval here is obtained as:

This is the required 95% confidence interval here.

c) From standard normal tables, we have here:
P(-2.576 < Z < 2.576) = 0.99

Therefore the confidence interval here is obtained as:

This is the required 99% confidence interval here.

Note that for the 2 intervals computed, both have the exact same mid point which is the point estimate of the difference in two proportions while the margin of error for 99% confidence interval is greater than that for the 95% confidence interval.