Final answer:
To estimate the mean age of movie patrons with 98% confidence and within 2 years of the actual mean with a standard deviation of 19.7 years, approximately 525 movie patrons need to be surveyed to satisfy the sample size. It is less likely that a sample from one movie at one theater provides a representative sample of the population.
Step-by-step explanation:
To find the sample size needed to estimate the population mean with a 98% confidence level and a margin of error of 2 years, we use the formula for the sample size of a mean which is:
n = (Z * σ / E)2
Where:
- n is the sample size,
- Z is the Z-score corresponding to the desired confidence level,
- σ (sigma) is the population standard deviation, and
- E is the margin of error.
Assuming σ = 19.7 years and a confidence level of 98%, we can find the corresponding Z-score (which is approximately 2.33 for 98% confidence). The calculation will be:
n = (2.33 * 19.7 / 2)2 = (45.791 / 2)2 = (22.8955)2 ≈ 524.44
Since we need a whole number for the sample size, we always round up, so we need approximately 525 movie patrons to achieve our desired level of confidence and precision.
As for whether the sample could be obtained from one movie at one theater, this scenario is less likely to provide a random and representative sample of the overall population since it may not reflect the diversity of movie patrons.