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Chapter 8: Problem 21

Consumption of alcoholic beverages by young women of drinking age has been increasing in the United Kingdom, the United States, and Europe (The Wall Street Journal, February 15,2006 ). Data (annual consumption in liters) consistent with the findings reported in The Wall Street Journal article are shown for a sample of 20 European young women. $$\begin{array}{crrrr} 266 & 82 & 199 & 174 & 97 \\ 170 & 222 & 115 & 130 & 169 \\ 164 & 102 & 113 & 171 & 0 \\ 93 & 0 & 93 & 110 & 1300 \end{array}$$ a. What is the \(95 \%\) confidence interval estimate for the mean ticket sales revenue per theater? Interpret this result. b. Using the movie ticket price of \(\$ 7.16\) per ticket, what is the estimate of the mean number of customers per theater? c. The movie was shown in 3118 theaters. Estimate the total number of customers who saw Hannah Montana: The Movie and the total box office ticket sales for the three- day weekend.

Short Answer

Expert verified
The 95% confidence interval for the mean annual alcohol consumption of European young women is calculated as \(\bar{x} \pm t^* \times SE\). Based on the given data and calculations, we are 95% confident that the true mean annual alcohol consumption lies within this range. Unfortunately, parts b and c contain mistakes in the question, as there is no direct connection between alcohol consumption and movie ticket sales.

Step by step solution

01

Calculate the Mean of the Sample

First, calculate the mean of the given dataset. The mean (\(\bar{x}\)) is the sum of all values divided by the number of values. Calculating the mean, we have: \[\bar{x} = \frac{266+82+199+174+97+170+222+115+130+169+164+102+113+171+0+93+0+93+110+1300}{20}\]
02

Calculate the Sample Standard Deviation

Now, calculate the standard deviation of the sample. The sample standard deviation (s) is calculated as: \[s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}\] Where \(x_i\) represents individual data points and n is the sample size.
03

Calculate the Standard Error

Next, calculate the standard error (SE). The standard error is calculated as: \[SE = \frac{s}{\sqrt{n}}\]
04

Find the T-Distribution Critical Value

For a 95% confidence interval, we need to find the t-distribution critical value (\(t^*\)) corresponding to \((1-0.95)/2 = 0.025\) in each tail with \(n-1 = 19\) degrees of freedom. You can find this value using a t-distribution table or software like Excel or R. In our case, \(t^* = 2.093\).
05

Calculate the 95% Confidence Interval

Now, calculate the confidence interval using the formula: \[\bar{x} \pm t^* \times SE\] Interpretation: The 95% confidence interval means that we are 95% confident that the true mean annual alcohol consumption of European young women lies within this range. For parts b and c, there is a mistake in the question as there is no direct connection between alcohol consumption and movie ticket sales. However, if you were given ticket sales data instead, you can use a similar process to compute the required information.

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Most popular questions from this chapter

According to statistics reported on \(\mathrm{CNBC}\), a surprising number of motor vehicles are not covered by insurance (CNBC, February 23,2006 ). Sample results, consistent with the CNBC report, showed 46 of 200 vehicles were not covered by insurance. a. What is the point estimate of the proportion of vehicles not covered by insurance? b. Develop a \(95 \%\) confidence interval for the population proportion.

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