Find the sample size needed to estimate the mean age of movie patrons such that it can be said with \( 98 \% \) confidence that the sample mean is within 1.5 years of the population mean. Assume that 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.)

To find the required sample size for estimating the mean age of movie patrons, we can use the formula for sample size determination when estimating a mean: \[ n = \left( \frac{Z_{\alpha/2} \cdot \sigma}{E} \right)^2 \] Where: - \( n \) is the required sample size, - \( Z_{\alpha/2} \) is the Z-score corresponding to the desired confidence level, - \( \sigma \) is the population standard deviation, - \( E \) is the margin of error. For a \( 98\% \) confidence level, the \( Z_{\alpha/2} \) value (from Z-tables) is approximately \( 2.33 \). The margin of error \( E \) is given as \( 1.5 \) years. However, we need the population standard deviation \( \sigma \). Assuming the previous report suggested a standard deviation of, say, \( 8 \) years (you will need to adjust this based on your data): 1. Plug in the numbers: \[ n = \left( \frac{2.33 \times 8}{1.5} \right)^2 \] \[ n = \left( \frac{18.64}{1.5} \right)^2 \] \[ n = \left( 12.43 \right)^2 \approx 154.55 \] 2. Round up to the nearest integer: \[ n = 155 \] So, the required sample size is \( 155 \). As for whether the sample should be obtained from one movie at one theater, it is generally not advisable to limit the sample to just one movie or one theater. This is because limiting your sample could introduce bias and not represent the overall population of movie patrons accurately. A wider selection of movies and theaters would provide a better estimate of the mean age.