A department store manager wants to estimate the mean amount...

A department store manager wants to estimate the mean amount spent by all customers at this store at a $98 \%$ confidence level. The manager knows that the standard deviation of amounts spent by all customers at this store is $\$ 31$. What minimum sample size should he choose so that the estimate is within $\$ 3$ of the population mean?

Key Concepts

Sample Size Calculation

Determining the sample size involves rearranging the margin of error formula so that the sample size is the subject of the equation. When the population standard deviation is known, the formula used is n = (Z * ? / E)^2, where ? is the population standard deviation, E is the margin of error, and Z is the Z-score corresponding to the desired confidence level. This formula guides the minimum sample size required to achieve a specific precision in the estimate.

Z-Score (Critical Value)

The Z-score, or critical value, corresponds to the desired confidence level in a standard normal distribution. It is used to scale the standard deviation in the calculation of the margin of error when the population standard deviation is known. For a 98% confidence level, the appropriate Z-score is used to capture the central 98% of the distribution.

Confidence Interval

A confidence interval provides a range of values which is likely to contain the population parameter of interest. In the context of estimating a mean, the interval is constructed around the sample mean and it represents the degree of uncertainty associated with estimating the true population mean. The confidence level (e.g., 98%) indicates how often the true mean would fall within the calculated intervals if the experiment were repeated multiple times.

Margin of Error

The margin of error is the maximum expected difference between the true population parameter and a sample estimate. It determines the amount of uncertainty in the estimate and ensures that the estimate is within a specified range (e.g., within $3 of the population mean). It directly influences the width of the confidence interval.

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