Final answer:
- a. The 99% confidence interval estimate for the mean ticket sales revenue per theater is approximately ($21,543 to $23,023).
- b. The estimate of the mean number of customers per theater is approximately 2,736 customers.
- c.The estimated total box office ticket sales for the weekend is approximately $91 million.
Step-by-step explanation:
To answer the questions, we need to analyze the given data and perform calculations based on it.
**a. To find the 95% confidence interval estimate for the mean ticket sales revenue per theater, we can use the formula: sample mean ± (critical value) × (standard deviation / sqrt(sample size)).
- 1. Calculate the sample mean: The sum of all revenue values divided by the number of theaters (30) gives us a sample mean of $22,783.
- 2. Calculate the standard deviation: First, find the sample variance by using the formula: sum of (data point - sample mean)^2 / (sample size - 1). After calculating the sample variance, take the square root to find the standard deviation, which is approximately $4,289.
- 3. Find the critical value: For a 95% confidence interval and a sample size of 30, the critical value is 2.045. This value can be found using a t-distribution table.
- 4. Plug in the values into the formula: $22,783 ± (2.045) × ($4,289 / √30). This gives us the 95% confidence interval estimate for the mean ticket sales revenue per theater: ($21,543 to $23,023).
Therefore, based on the given data, the 95% confidence interval estimate for the mean ticket sales revenue per theater is approximately ($21,543 to $23,023).
**b. To estimate the mean number of customers per theater, we can divide the total ticket sales revenue by the movie ticket price. Given that the ticket price is $8.16, we divide the mean ticket sales revenue per theater ($22,356.13) by $8.16:
- Mean number of customers per theater ≈ $22,356.13 / $8.16
Rounding to the nearest whole number, the estimate of the mean number of customers per theater is approximately 2,736 customers.
**c. To estimate the total number of customers who saw Guardians of the Galaxy and the total box office ticket sales for the weekend, we need to multiply the mean number of customers per theater by the number of theaters.
- Total number of customers ≈ Mean number of customers per theater * Number of theaters
- Total number of customers ≈ 2,736 * 4,070
The estimated total number of customers who saw Guardians of the Galaxy is approximately 11,137,920 customers.
To estimate the total box office ticket sales, we multiply the mean ticket sales revenue per theater by the number of theaters:
- Total box office ticket sales ≈ Mean ticket sales revenue per theater * Number of theaters
- Total box office ticket sales ≈ $22,356.13 * 4,070
Rounding to the nearest million, the estimated total box office ticket sales for the weekend is approximately $91 million.