Ages of Moviegoers Find the sample size needed to estimate the mean age of movie patrons, given that we want 98% confidence that the sample mean is within 1.5 years of the population mean. Assume that sigma=19.6 years, based on a previous report from the Motion Picture Association of America. Could the sample be obtained from one movie at one theater?

Verified step by step guidance

1. Identify the formula for determining the required sample size for estimating a population mean: $n\; =\; (z\; *\; \sigma \; /\; E)^2$, where $n$ is the sample size, $z$ is the z-score corresponding to the desired confidence level, $\sigma$ is the population standard deviation, and $E$ is the margin of error.
2. Determine the z-score for a 98% confidence level. For a two-tailed test, divide the remaining 2% (1 - 0.98) equally into the two tails, leaving 1% in each tail. Use a z-table or statistical software to find the z-score corresponding to 0.99 (cumulative probability).
3. Substitute the given values into the formula: $\sigma \; =\; 19.6$ (population standard deviation) and $E\; =\; 1.5$ (margin of error). The formula becomes $n\; =\; (z\; *\; 19.6\; /\; 1.5)^2$.
4. Simplify the expression by first calculating $z\; *\; 19.6$, then dividing by $1.5$, and finally squaring the result to find $n$.
5. Consider whether the sample could be obtained from one movie at one theater. Reflect on whether the sample would be representative of the entire population of moviegoers, as a single theater might introduce bias due to location, demographics, or other factors.

Key Concepts

Sample Size Calculation

Sample size calculation is a statistical method used to determine the number of observations or replicates needed in a study to achieve a desired level of precision. In this context, it involves using the formula that incorporates the desired confidence level, the population standard deviation, and the margin of error to estimate the mean age of moviegoers accurately.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. In this case, a 98% confidence level indicates that if we were to take many samples, approximately 98% of the calculated intervals would contain the true mean age of movie patrons.

Population Standard Deviation

Population standard deviation is a measure of the amount of variation or dispersion in a set of values. It is denoted by sigma (σ) and is crucial in sample size calculations, as it reflects how much individual ages of moviegoers deviate from the mean age. A known standard deviation allows for more accurate estimations of the required sample size.