Questions: 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 σ=19.8 years, based on a previous report. Should the sample be obtained from one movie at one theater? The required sample size is (Round up to the nearest integer.)

Solution Steps

To find the required sample size, we first need the z-score corresponding to a \(98\%\) confidence level. This is calculated as:

\[ z = z_{\alpha/2} = z_{0.01} \approx 2.3263 \]

The formula for the sample size \(n\) needed to estimate the mean is given by:

\[ n = \left( \frac{z \cdot \sigma}{E} \right)^2 \]

where:

• \(z\) is the z-score,
• \(\sigma\) is the population standard deviation,
• \(E\) is the margin of error.

Substituting the known values:

\[ n = \left( \frac{2.3263 \cdot 19.8}{1.5} \right)^2 \]

Calculating the expression:

\[ n = \left( \frac{46.073934}{1.5} \right)^2 \approx \left(30.715956\right)^2 \approx 943.0000 \]

Since the sample size must be a whole number, we round up:

\[ n = \lceil 943.0000 \rceil = 943 \]

Final Answer