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A sample of 20 pages was taken from a Yellow Pages (yp.com) business directory. On each page, the mean area devoted to display ads was measured. The data (in square millimeters) is shown below:

[tex]
\[
\begin{tabular}{rrrrrrrrrr}
0 & 260 & 356 & 403 & 536 & 0 & 268 & 369 & 428 & 536 \\
268 & 396 & 469 & 536 & 162 & 338 & 403 & 536 & 536 & 130
\end{tabular}
\]
[/tex]

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

Note: Round your answers to 2 decimal places.

The 95 percent confidence interval is from [tex]346.50[/tex] to [tex]170.38[/tex].

(b) What sample size would be needed to obtain an error of [tex]\pm 20[/tex] square millimeters with 95 percent confidence?

Note: Enter your answer as a whole number (no decimals). Use a z-value taken to 3 decimal places in your calculations.

Sample size: [tex]\square[/tex]

Answer :

a. The 95 percent confidence interval for the true mean is from 368.17 to 495.73.

b. A sample size of 180 is needed.

a.The sample mean is the average of the 20 data points.

Sum of values:

0 + 260 + 356 + 403 + 536 + 0 + 268 + 369 + 428 + 536 + 268 + 396 + 469 + 536 + 162 + 338 + 403 + 536 + 536 + 130 = 8639

Sample mean:

= 8639 / 20 = 431.95

Calculate the sample standard deviation (s):

The sample standard deviation is calculated using the formula:

s = sqrt[ Σ(xi - x)² / (n - 1) ]

Using the data, the sample standard deviation (s) is approximately 136.11.

For a sample size of 20 (degrees of freedom = 19), the t-value for a 95% confidence level is approximately 2.093.

Calculate the margin of error (ME):

The margin of error is calculated as:

ME = t * (s / √n)

ME = 2.093 * (136.11 / √20) ≈ 63.78

Calculate the confidence interval:

The confidence interval is:

CI = (x - ME, x + ME)

CI = (431.95 - 63.78, 431.95 + 63.78)

CI = (368.17, 495.73)

(b) Desired margin of error (E) = 20

Confidence level = 95% (which corresponds to a z-value of 1.960)

The formula for the sample size needed is:

n = (z * s / E)²

Substituting the values:

z = 1.960

s = 136.11

E = 20

n = (1.960 * 136.11 / 20)²

n = (267.60 / 20)²

n = (13.38)²

n ≈ 179.12

Rounding up, the required sample size is 180.

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