3. Playbill magazine reported that the mean annualhousehold income of its reader

Question

3. Playbill magazine reported that the mean annualhousehold income of its
readers is $119,155. Assuming that this estimate of the mean annualhousehold
income is based on a sample of 80 households, and based on paststudies, the
population standard deviation is known to be $30,000.

a. Develop 90%, 95%, and 99% confidence interval estimates ofthe population
mean.
b. Discuss what happens to the width of the confidenceinterval as the
confidence level is increased. Does this result seem reasonable?Explain.

Explanation / Answer

Given xbar= 119155, s=30000, n=80 (a) =0.1, Z(0.05)=1.645 (check normal table) The 90% CI is xbar ± Z*s/n --> 119155 ± 1.645*30000/sqrt(80) --> (113637.5, 124672.5) ------------------------------------------------------------- =0.05, Z(0.025)=1.96 (check normal table) xbar ± Z*s/n --> 119155 ± 1.96*30000/sqrt(80) --> (112581, 125729) ------------------------------------------------------------- =0.01, Z(0.005)=2.58 (check normal table) xbar ± Z*s/n --> 119155 ± 2.58*30000/sqrt(80) --> (110501.4, 127808.6) ------------------------------------------------------------- (b)the width of the confidence interval increases as the confidence level is increased

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