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.)
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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.