A researcher wishes to estimate the average number of...

A researcher wishes to estimate the average number of minutes per day a person spends on the Internet. How large a sample must she select if she wishes to be $90 \%$ confident that the population mean is within 10 minutes of the sample mean? Assume the population standard deviation is 42 minutes.

Key Concepts

Confidence Interval

A confidence interval is a range derived from sample data that is likely to contain the true population parameter. It is constructed from the sample mean and the standard error, reflecting the uncertainty inherent in the estimate.

Margin of Error

The margin of error represents the maximum expected difference between the sample mean and the true population mean. It is determined by the product of the critical value (from the confidence level) and the standard error, outlining the precision of the estimate.

Population Standard Deviation

The population standard deviation is a measure of the spread or dispersion of data in the entire population. When known, it is used to calculate the standard error of the mean, which in turn is crucial for constructing confidence intervals and determining the required sample size.

Critical Value (Z-Score)

The critical value, often a Z-score in cases where the population standard deviation is known, corresponds to the desired confidence level in a standard normal distribution. It scales the standard error to capture the required proportion of the distribution for the confidence interval.

Sample Size Calculation

Sample size calculation is the process of determining the minimum number of observations needed to estimate a population parameter within a desired margin of error at a specified confidence level. The calculation typically involves rearranging the confidence interval formula to solve for the sample size using the critical value, population standard deviation, and margin of error.

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