Disney's Hannah Montana: The Movie opened on Easter weekend in April 2009. Over the three-day weekend, the movie became the number-one box office attraction (The Wall Street Journal, April 13,2009 ). The ticket sales revenue in dollars for a sample of 25 theaters is as follows. $$\begin{array}{rrrrr} 20,200 & 10,150 & 13,000 & 11,320 & 9,700 \\ 8,350 & 7,300 & 14,000 & 9,940 & 11,200 \\ 10,750 & 6,240 & 12,700 & 7,430 & 13,500 \\ 13,900 & 4,200 & 6,750 & 6,700 & 9,330 \\ 13,185 & 9,200 & 21,400 & 11,380 & 10,800 \end{array}$$ 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 threeday weekend.

The sample mean, denoted as \( \bar{x} \), is a measure of the central tendency of a set of data. It represents the average value in a sample and is calculated by summing all the numbers in the sample and then dividing by the quantity of numbers.

In the context of the Hannah Montana movie revenue example, the sample mean of ticket sales revenue was computed by adding the revenue from each of the 25 theaters and dividing by 25. This gave us the average ticket sales revenue per theater. Understanding the calculation of the sample mean is crucial, as it is a building block for many statistical measures and is used directly in the computation of the confidence interval in further steps of the exercise.

Knowing how to calculate and interpret the sample mean allows students to ascertain the general performance of a group, such as the average earnings from ticket sales across a selection of theaters in a given time period.

Standard deviation is a statistic that measures the amount of variability or dispersion around the mean of a set of values. It indicates how spread out the numbers in your data are, with a high standard deviation signifying that the data points are widely scattered from the mean, and a low standard deviation indicating that they are closely clustered.

To calculate the standard deviation, which is symbolized as \( s \), you subtract the sample mean from each data point, square the result to make it positive, average these squared differences by dividing by the number of data points minus 1 (to account for sample data, known as degrees of freedom), and finally take the square root of this average. In our exercise, the standard deviation of ticket sales revenue helps us understand the consistency of revenue across different theaters.

Note: When dealing with a sample rather than an entire population, we use 's' to represent the standard deviation and we divide by \( n-1 \) (degrees of freedom) rather than \( n \) when calculating the variance (which is the square of the standard deviation).

The t-distribution is a type of probability distribution that is symmetric and bell-shaped, like the normal distribution, but it has heavier tails. This means that it predicts a greater number of extreme values than the normal distribution. It is particularly useful when dealing with small sample sizes or when the population standard deviation is unknown, both of which apply to the Hannah Montana exercise scenario.

When constructing a confidence interval estimate with a t-distribution, we use the t-score, which adjusts for the size of the sample. If you have a larger sample size, the t-distribution approaches the normal distribution. The relevant formula for the confidence interval using the t-distribution is:
\[ \bar{x} \pm t_{\alpha/2, df} \cdot \frac{s}{\sqrt{n}} \]
where \( t_{\alpha/2, df} \) is the t-score that corresponds to the desired confidence level and degrees of freedom (df), which is \( n-1 \). In our movie theater example, the t-distribution is used to estimate the mean ticket sales revenue with a 95% confidence interval, reflecting the uncertainty due to the small sample size.