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Chapter 8: Problem 6

Nielsen Media Research conducted a study of household television viewing times during the 8 P.M. to 11 P.M. time period. The data contained in the file named Nielsen are consistent with the findings reported (The World Almanac, 2003). Based upon past studies, the population standard deviation is assumed known with \(\sigma=3.5\) hours. Develop a \(95 \%\) confidence interval estimate of the mean television viewing time per week during the 8 P.M. to 11 P.M. time period.

Short Answer

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To find the 95% confidence interval estimate for the mean television viewing time per week during the 8 P.M. to 11 P.M. time period, first calculate the sample mean \(\bar{x}\) from the Nielsen data file. Next, use the Z-score of 1.96 for a 95% confidence interval, and the given population standard deviation \(\sigma = 3.5\) hours. Compute the margin of error as follows: \(ME = 1.96 \times \frac{3.5}{\sqrt{n}}\), where \(n\) is the sample size. Finally, calculate the 95% confidence interval: \(CI = (\bar{x} - ME, \bar{x} + ME)\). This will give you a range representing the lower and upper bounds of the estimated mean television viewing time per week during the specified time period.

Step by step solution

01

Calculate the sample mean \(\bar{x}\)

For this, you will have to go through the given file named "Nielsen" and calculate the mean of the data points. We will denote this sample mean by \(\bar{x}\).
02

Find the Z-score for a 95% confidence interval

Since we are constructing a 95% confidence interval, we want to find the critical value (Z-score) to use in the calculations. The Z-score for a 95% confidence interval is approximately 1.96. This value corresponds to the point where 95% of the area under the standard normal distribution curve is captured.
03

Calculate the margin of error

Now that you have the sample mean, \(\bar{x}\), the population standard deviation, \(\sigma\), and the Z-score, you can calculate the margin of error (ME) as follows: \[ ME = Z \times \dfrac{\sigma}{\sqrt{n}} \] where \(n\) represents the sample size.
04

Calculate the 95% confidence interval

Finally, you can calculate the 95% confidence interval by adding and subtracting the margin of error from the sample mean: \[ CI = (\bar{x} - ME, \bar{x} + ME) \] Your answer will be in the form of a range representing the lower and upper bounds of the 95% confidence interval for the mean television viewing time per week between 8 P.M. and 11 P.M.

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