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Find the sample sise needed to estimate the mean age of movie patrons such that it can be said with \( 98 \% \) contidence that the sample mean is within 1.5 years of the based on a previous report. Should the sample be obtained from one movie at one theater? The required sample size is \( \square \). Round up to the nearest integer.)

Ask by Adkins Kelly. in the United States
Mar 15,2025

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The required sample size is 93.

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To estimate the mean age of movie patrons with a 98% confidence level and a margin of error of 1.5 years, we first need the standard deviation (σ) from the previous report. Assuming σ is approximately 10 years for this example, we can use the formula for sample size \( n \): \[ n = \left( \frac{Z \times \sigma}{E} \right)^2 \] Where \( Z \) is the z-score for 98%, which is approximately 2.33, and \( E \) is the margin of error (1.5 years). Plugging in the values: \[ n = \left( \frac{2.33 \times 10}{1.5} \right)^2 \approx 88.67 \] Rounding up, you would need a sample size of \( 89 \) patrons. For a representative sample, it's better to gather data from multiple screenings and theaters rather than just one movie in one theater. This accounts for variances in patron demographics and provides a broader perspective of the overall audience age.

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