Question

Does age make a difference in the amount of savings a worker feels is needed to be secure at retirement? A study by CommSciences for Transamerica Asset Manage- ment found that 0.24 of workers in the 25–33 age category feel that $250,000 to $500,000 is enough to be secure at retirement. However, 0.35 of the workers in the 34–52 age category feel that this amount is enough. Suppose 210 workers in the 25–33 age category and 176 workers in the 34–52 age category were involved in this study. Use these data to construct a 90% confidence interval to estimate the differ- ence in population proportions on this question.

A. Show all the steps and in words give an interpretation of the resulting interval.

B. A 95% confidence interval for the ages of the first five Academic Deans at the time they were appointed the position was 50.36 to 77.78. Either interpret the interval or explain why it should not be interpreted.

Answer 1

A.

Let p1 , p2 be the true proportion of workers in the 25–33 age category and 34–52 age category respectively who feel that $250,000 to $500,000 is enough to be secure at retirement.

Point Estimate of the difference in proportions, p2 - p1 = 0.35 - 0.24 = 0.11

Pooled Proportion, p = (n1 p1 + n2 p2) / (n1 + n2) = (210 * 0.24 + 176 * 0.35) / (210 + 176) = 0.2901554

Standard Error of proportion, SE = $\sqrt{p(1-p)((1/n_1)+(1/n_2))} = \sqrt{0.2901554(1-0.2901554)((1/210)+(1/176))}$

= 0.04637935

Z value for 90% confidence interval is 1.645

90% confidence interval of the difference in proportion of workers in the 25–33 age category and 34–52 age category respectively who feel that $250,000 to $500,000 is enough to be secure at retirement is,

(0.11 - 1.645 * 0.04637935 ,  0.11 + 1.645 * 0.04637935)

(0.0337 ,  0.1863)

B.

We're 95% confident that the interval (50.36, 77.78) captured the true mean ages of the first five Academic Deans at the time they were appointed the position.