Movie Ticket Sales. Marvel Studio's motion picture Guardians...

Movie Ticket Sales. Marvel Studio's motion picture Guardians of the Galaxy opened over the first two days of the 2014 Labor Day weekend to a then recordbreaking $$\$ 94.3$$ million in ticket sales revenue in North America. The ticket sales revenue in dollars for a sample of 30 theaters during the same dates is as follows. 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 $$\$ 8.11$$ per ticket, what is the estimate of the mean number of customers per theater? c. The movie was shown in 4080 theaters. Estimate the total number of customers who saw Guardians of the Galaxy and the total box office ticket sales for the weekend.

Movie Ticket Sales. Marvel Studio's motion picture Guardians of the Galaxy opened over the first two days of the 2014 Labor Day weekend to a then recordbreaking $$\$ 94.3$$ million in ticket sales revenue in North America. The ticket sales revenue in dollars for a sample of 30 theaters during the same dates is as follows.
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 $$\$ 8.11$$ per ticket, what is the estimate of the mean number of customers per theater?
c. The movie was shown in 4080 theaters. Estimate the total number of customers who saw Guardians of the Galaxy and the total box office ticket sales for the weekend.

Key Concepts

Extrapolation Using Sample Estimates

Extrapolation is the process of extending conclusions from the sample to the broader population. By applying the sampled estimates—such as mean revenue or number of customers—to a larger number of units (in this case, theaters), one can estimate overall totals. This method relies on the assumption that the sample accurately represents the population and that the relationships observed in the sample continue uniformly across the population.

Data Transformation in Statistical Estimation

Often, it is necessary to transform estimates from one measure to another through mathematical operations, such as scaling. In this context, converting ticket sales revenue into the number of tickets sold (or customers) involves dividing by a constant unit price. This linear transformation preserves the estimation properties while translating the result into a more directly interpretable unit.

Confidence Interval Estimation

This concept involves using sample data to calculate a range of values that are likely to contain the true population mean. When the sample size is relatively small and the population standard deviation is unknown, the t-distribution is used instead of the normal distribution, leading to a confidence interval of the form sample mean ± (t* × standard error). This method quantifies the uncertainty inherent in sampling and provides a probabilistic range for where the true mean is expected to lie.

Interpretation of Confidence Interval

Interpreting a confidence interval means understanding that if the sampling process were repeated many times, a certain percentage (in this case 95%) of the constructed intervals would capture the true population mean. It is a statement about the reliability of the estimation process rather than a probability statement about the population parameter itself.

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