Help me I don't have a clue / Applied Statistics in Business and Economics?

Question

A sample of 20 pages was taken without replacement from the 1,591-page phone directory Ameritech Pages Plus Yellow Pages. On each page, the mean area devoted to display ads was measured (a display ad is a large block of multicolored illustrations, maps, and text). The data (in square millimeters) are shown below:

0 260 356 403 536 0 268 369 428 536

268 396 469 536 162 338 403 536 536 130

(a) Construct a 95 percent confidence interval for the true mean.

(b) Why might normality be an issue here?

(c) What sample size would be needed to obtain an error of ±10 square millimeters

with 99 percent confidence?

(d) If this is not a reasonable requirement, suggest one that is. (Data are from a project by MBA student Daniel R. Dalach.)

Answer 1

(a) ANSWER: 95% CONFIDENCE INTERVAL = 266.8 sq-mm, 426.2 sq-mm]

Why??

SMALL SAMPLE, CONFIDENCE INTERVAL, NORMAL POPULATION DISTRIBUTION

x-bar = Sample mean [346.5]

s = Sample standard deviation [170.4]

n = Number of samples [20]

df = degrees of freedom [19] (n - 1)

For confidence level of 95%, two-sided interval ("look-up" from Table) "t critical value" [2.093]

Resulting Confidence Interval for "true mean":

x-bar +/- (t critical value) * s/SQRT(n) = 346.5 +/- 2.093 * 170.4/SQRT(20) = [266.8 sq-mm, 426.2 sq-mm]

(b) ANSWER: Finite limit of sq-mm per page not absolutely normal.  e.g. theoretically infinite +/- range limit of sq-mm per page not possible.  Rather; acceptably close approximation of normal distribution utilized for purposes of estimating  confidence interval.

(c) ANSWER: SAMPLE SIZE = 1927 PAGES for 99% CONFIDENCE OF +/- 10 sq-mm

Why??

CHOOSING THE SAMPLE SIZE

n = [(z-critical value * s)/B]^2

s = STANDARD DEVIATION [170.4]

B = BOUND ON THE ERROR OF ESTIMATION [10]

z - critical value ASSOCIATED WITH 99% ("two-sided") CONFIDENCE [2.576]

n = [(2.576 * 170.4)/10]^2 = 1927

CONCLUSION: A random selection of 1927 pages (larger than Ameritech Pages Plus Yellow Pages) will provide 99% confidence estimating the true mean area devoted to display adds.

(d) ANSWER: "Reasonable" statistical research in alloted time given by the boss.  Advertising Manager of Yellow Pages might ostensibly assign "summer intern" the task of surveying past years' and competitors' Yellow Pages to learn how much area devoted to advertising.  Such a task might be hours to several days in time duration.