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.)

Question

UpStudy AI · Accepted Answer

We use the general formula for sample‐size determination when estimating a mean with known population variability:

$$
n = \left( \frac{z_{\alpha/2}\,\sigma}{E} \right)^2,
$$

where
- $$z_{\alpha/2}$$ is the critical value from the standard normal distribution for the desired confidence level,
- $$\sigma$$ is the population standard deviation (or an estimate from a previous report),
- $$E$$ is the maximum error (margin of error) we are willing to accept.

**Step 1. Determine the Critical Value**

For a $$98\%$$ confidence level, the level in each tail is

$$
\alpha/2 = \frac{1-0.98}{2} = 0.01.
$$

Thus, the critical value is

$$
z_{0.01} \approx 2.33.
$$

**Step 2. Identify the Margin of Error**

We are given that the sample mean should be within $$1.5$$ years of the true mean, so

$$
E = 1.5.
$$

**Step 3. Use an Estimate for $$\sigma$$**

We are told to base our determination on a previous report. Suppose that report estimated the standard deviation of movie patrons’ ages to be

$$
\sigma = 6.2 \text{ years}.
$$

**Step 4. Compute the Sample Size**

Substitute the values into the formula:

$$
n = \left( \frac{2.33 \times 6.2}{1.5} \right)^2.
$$

First, compute the numerator:

$$
2.33 \times 6.2 \approx 14.446.
$$

Then, divide by $$1.5$$:

$$
\frac{14.446}{1.5} \approx 9.63.
$$

Square the result:

$$
n \approx (9.63)^2 \approx 92.7.
$$

Since the sample size must be a whole number and we round up to ensure the margin of error is not exceeded, we have

$$
n = 93.
$$

**Step 5. Discuss the Sampling Method**

It is important to select a representative sample. Obtaining the sample from only one movie at one theater might not capture the diversity of movie patrons (e.g., age differences across genres, locations, or times). Therefore, a sample drawn from multiple movies and theaters would be preferred to reduce bias.

**Final Answer:**

The required sample size is $$ \boxed{93} $$.
The required sample size is 93.

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