Confidence intervals: AE - 10

Match each item on the left with its correct description on the right.

1.\(y \pm 2se(\bar{y})\) (a)  Sampling distribution of X

2. \(\hat{p} \pm se(\hat{p})\) (b)  Margin of error

3. \(2se (\bar{X})\) (c)  100% confidence interval for p

4. \(N(\mu,\sigma^2/n)\) (d)  Estimated standard error of \(\bar{Y}\)

5. \(s/\sqrt{n}\) (e)  Estimated standard error of \(\hat{p}\)

6. \(\sigma/\sqrt{n}\) (f)  An interval with about 95% coverage

7. \(1/(0.05)^2\) (g)  Actual standard error of \(\bar{Y}\)

8. [0, 1] (h) About 2 for moderate sample sizes

9. \(\sqrt{\hat{p}(1 -\hat{p})/n}\) (i)   An interval with 68% coverage

10. \(t_{0.025, n-1}\) (j)   Sample size needed for 0.05 margin of error

11. True or False: By increasing the sample size from \(n = 100\) to \(n = 400\), we can reduce-the margin of error by \(50\%\).

12. If the 95% confidence interval for the average purchase of customers at a department store is $50 to $110, then $100 is a plausible value for the population mean at this level of confidence.

13. If zero lies inside the 95% confidence interval for \(\mu\), then zero is also inside the 99% confidence interval for \(\mu\).

14. The clothing buyer for a department store wants to order the right mix of sizes. As part of a survey, she measured the height (in inches) of men who bought suits at this store. Her software reported the following confidence interval:

With 95.00% confidence, 70.8876 < \(\mu\) < 74.4970

(a) Explain carefully what the software output means.

(b) What’s the margin of error for this interval?

(c) How should the buyer round the endpoints of the interval to summarize the result in a report for store managers?

(d) If the researcher had calculated a 99% confidence interval, would the output have shown a longer or shorter interval?

15. Hoping to lure more shoppers downtown, a city builds a new public parking garage in the central business district. The city plans to pay for the structure through parking fees. During a two-month period (44 weekdays), daily fees collected averaged $1,264 with a standard deviation of $150.

(a) What assumptions must you make in order to use these statistics for inference?

(b) Write a 90% confidence interval for the mean daily income this parking garage will generate, rounded appropriately.

(c) The consultant who advised the city on this project predicted that parking revenues would average $1,300 per day. On the basis of your confidence interval, do you think the consultant was correct? Why or why not?

(d) Give a 90% confidence interval for the total revenue earned during five weekdays.