Question

In a survey of 1006 adults, a polling agency asked, "When you retire, do you think you will have enough money to live comfortably or not?" Of the 1006 surveyed, 530 stated that they were worried about having enough money to live comfortably in retirement. Construct a 90% confidence interval for the proportion of adults who are worried about having enough money to live comfortably in retirement.

a) 52.5% to 55.9%
b) 50.6% to 54.2%
c) 51.2% to 55.8%
d) 53.0% to 56.4%

Answer 1

Construct a 90% confidence interval for the proportion of adults who are worried about having enough money to live comfortably in retirement is 51.2% to 55.8%. Thus, the correct answer is option c) 51.2% to 55.8%.

Step-by-step explanation:

To construct a 90% confidence interval for the proportion of adults worried about having enough money in retirement, we can use the formula for a confidence interval for a proportion:

`\[ \text{Confidence Interval} = \hat{p} \pm Z * \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]`

where

-
`\(\hat{p}\)` is the sample proportion (worrying about having enough money),

- Z is the z-score corresponding to the desired confidence level,

- n is the sample size.

In this case,
`\(\hat{p} = (530)/(1006) \approx 0.5274\), \(n = 1006\)`, and for a 90% confidence interval, the z-score is approximately 1.645.

Now, plug in these values into the formula:

`\[ \text{Confidence Interval} = 0.5274 \pm 1.645 * \sqrt{(0.5274 * (1-0.5274))/(1006)} \]`

Calculating this gives the confidence interval:

`\[ \text{Confidence Interval} \approx (0.512, 0.558) \]`

Therefore, the correct answer is c) 51.2% to 55.8%. This means we are 90% confident that the true proportion of adults worried about having enough money to live comfortably in retirement lies between 51.2% and 55.8%.