A researcher wanted to determine the mean number of hours...

A researcher wanted to determine the mean number of hours per week (Sunday through Saturday) the typical person watches television. Results from the Sullivan Statistics Survey I indicate that $s=7.5$ hours. (a) How many people are needed to estimate the number of hours people watch television per week within 2 hours with $95 \%$ confidence? (b) How many people are needed to estimate the number of hours people watch television per week within 1 hour with $95 \%$ confidence? (c) What effect does doubling the required accuracy have on the sample size? (d) How many people are needed to estimate the number of hours people watch television per week within 2 hours with $90 \%$ confidence? Compare this result to part (a). How does decreasing the level of confidence in the estimate affect sample size? Why is this reasonable?

A researcher wanted to determine the mean number of hours per week (Sunday through Saturday) the typical person watches television. Results from the Sullivan Statistics Survey I indicate that $s=7.5$ hours.
(a) How many people are needed to estimate the number of hours people watch television per week within 2 hours with $95 \%$ confidence?
(b) How many people are needed to estimate the number of hours people watch television per week within 1 hour with $95 \%$ confidence?
(c) What effect does doubling the required accuracy have on the sample size?
(d) How many people are needed to estimate the number of hours people watch television per week within 2 hours with $90 \%$ confidence? Compare this result to part (a). How does decreasing the level of confidence in the estimate affect sample size? Why is this reasonable?

Key Concepts

Confidence Interval

A confidence interval provides a range of values that is likely to contain the true population parameter. It is constructed from the sample statistic plus and minus a margin of error, which is based on the variability of the data and the desired level of confidence.

Margin of Error

The margin of error is the amount by which the sample estimate is expected to differ from the true population parameter. It is calculated as the product of a critical value (which depends on the confidence level) and the standard error, and it establishes the precision of the estimate.

Critical Value and Confidence Level

The critical value is derived from the probability distribution (typically the normal distribution when the population standard deviation is known) and corresponds to the desired confidence level. A higher confidence level increases the critical value, leading to a wider interval and requiring a larger sample size for the same precision.

Sample Size Determination

Determining the sample size involves rearranging the margin of error formula so that the sample size is expressed in terms of the desired margin of error, the standard deviation, and the critical value. This calculation shows that reducing the margin of error or increasing the confidence level generally requires a larger sample size.

Impact of Desired Accuracy

Improving the accuracy of an estimate, which means reducing the margin of error, increases the required sample size significantly. Specifically, if the margin of error is halved, the sample size must be quadrupled, demonstrating the quadratic relationship between precision and the sample size.

Effect of Changing Confidence Level

Adjusting the confidence level affects the critical value used in the margin of error calculation. Lowering the confidence level reduces the critical value, which in turn decreases the required sample size for a given margin of error. This trade-off is reasonable since there is less certainty about capturing the true parameter, but it allows for a more economical study design.

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