Customers arrive at a movie theater at the advertised movie time only to find that they have to sit through several previews and prepreview ads before the movie starts. Many complain that the time devoted to previews is too long (The Wall Street Journal, October 12, 2012). A preliminary sample conducted by The Wall Street Journal showed that the standard deviation of the amount of time devoted to previews was four minutes. Use that as a planning value for the standard deviation in answering the following questions.
a. If we want to estimate the population mean time for previews at movie theaters with a margin of error of 75 seconds, what sample size should be used? Assume 95% confidence.
b. If we want to estimate the population mean time for previews at movie theaters with a margin of error of 1 minute, what sample size should be used? Assume 95% confidence.
a) 40
b) 62
To solve these questions, we will use the formula for sample size estimation:
n = (Z * σ / E)^2
Where:
n = Sample size
Z = Z-score (corresponding to the desired confidence level)
σ = Standard deviation
E = Margin of error
a. To estimate the population mean time for previews at movie theaters with a margin of error of 75 seconds and a 95% confidence level:
Z = Z-score for a 95% confidence level, which corresponds to a value of 1.96.
σ = Standard deviation of the amount of time devoted to previews = 4 minutes = 240 seconds
E = Margin of error = 75 seconds
Substituting the values into the formula:
n = (1.96 * 240 / 75)^2
n ≈ 6.2792^2
n ≈ 39.4116
Therefore, a sample size of approximately 40 should be used.
b. To estimate the population mean time for previews at movie theaters with a margin of error of 1 minute and a 95% confidence level:
Z = Z-score for a 95% confidence level, which is 1.96
σ = Standard deviation of the amount of time devoted to previews = 4 minutes = 240 seconds
E = Margin of error = 1 minute = 60 seconds
Substituting the values into the formula:
n = (1.96 * 240 / 60)^2
n ≈ 7.84^2
n ≈ 61.4656
Therefore, a sample size of 62 should be used.
To estimate the sample size required to estimate the population mean time for previews at movie theaters with a specific margin of error, we can use the formula for sample size calculation:
n = (Z * σ / E)^2
where:
- n is the sample size needed
- Z is the z-score corresponding to the desired level of confidence (in this case, 95% confidence corresponds to a z-score of approximately 1.96)
- σ is the standard deviation
- E is the margin of error
a. To estimate the population mean time for previews at movie theaters with a margin of error of 75 seconds and assuming a standard deviation of 4 minutes (or 240 seconds), we can calculate the sample size as follows:
n = (1.96 * 240 / 75)^2
n = 6.24^2
n = 38.81
Therefore, a sample size of 39 should be used.
b. To estimate the population mean time for previews at movie theaters with a margin of error of 1 minute (or 60 seconds) and assuming a standard deviation of 4 minutes (or 240 seconds), we can calculate the sample size as follows:
n = (1.96 * 240 / 60)^2
n = 7.84^2
n = 61.46
Therefore, a sample size of 62 should be used.