Playbill magazine reported that the mean annual household income of its readers is \(\$ 119,155(\text {Playbill}, \text { January } 2006) .\) Assume this estimate of the mean annual household income is based on a sample of 80 households, and, based on past studies, the population standard deviation is known to be \(\sigma=\$ 30,000\) a. Develop a \(90 \%\) confidence interval estimate of the population mean. b. Develop a \(95 \%\) confidence interval estimate of the population mean. c. Develop a \(99 \%\) confidence interval estimate of the population mean. d. Discuss what happens to the width of the confidence interval as the confidence level is increased. Does this result seem reasonable? Explain.

The population mean is an essential concept in statistics, representing the average value in a entire population. When you see references to mean household income, like in the Playbill magazine case (\$119,155), it's often an estimate derived from a sample. The true population mean is rarely known because it's impractical to survey every household; hence we use samples to estimate it. By analyzing a small slice of the population, a sample mean \(\bar{x}\) can provide a good estimation of the population mean; however, to ensure our estimate is reliable, we make use of confidence intervals.

Confidence intervals allow us to state with a certain level of assurance, or 'confidence', how close our sample mean is to the true population mean. This concept is pivotal because it accounts for variability within a sample and provides a range that's likely to contain the population mean rather than giving a single number that might be misleading.

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. If you're familiar with the concept of the 'bell curve' or normal distribution, the Z-score helps you find where a particular value lies on that curve.

In the context of confidence intervals, the Z-score lets us determine how 'far' we need to go from the sample mean to achieve a certain level of confidence. For instance, a 90% confidence level corresponds to a Z-score of approximately 1.645. This means we're looking for a range that extends 1.645 standard deviations on each side of the sample mean, which gives us our confidence interval. Higher confidence levels like 95% and 99% correspond to Z-scores of 1.960 and 2.576, respectively, implying a wider spread due to increased certainty.

Standard deviation (often represented as \(\sigma\) when referring to the population and \(s\) for a sample) is a measure of the amount of variation or dispersion of a set of values. A low standard deviation means that the values tend to be close to the mean, whereas a high standard deviation indicates that the values are spread out over a wider range.

In our Playbill magazine example, the population standard deviation of household income is reported as \$30,000. This figure is crucial when calculating the margin of error for confidence intervals, as it directly influences how wide the interval will be. When the standard deviation is high, it suggests there's more variability in the data, and consequently, our confidence interval will need to be wider to account for this variability and provide a reliable estimate.

Sample size, denoted as \(n\), plays a direct role in determining the precision of our confidence interval estimates. The sample size indicates the number of observations from the population used to estimate the population mean. In the context of the Playbill magazine's income report, the sample size is 80 households.

Larger sample sizes tend to yield more precise estimates of the population mean, as they reduce the margin of error in the confidence interval. This is due to the inverse relationship between sample size and the standard error (the standard deviation of the sampling distribution of a statistic, such as the mean), which means as you increase the number of observations, the variability of your estimate decreases. It's like getting more opinions before making a decision—the more you have, the better informed you are likely to be.

The confidence level is a value that indicates how sure we can be in the process that generates the confidence intervals. It is expressed as a percentage, such as 90%, 95%, or 99%. Higher confidence levels imply greater certainty in the reliability of the interval but also result in broader ranges.

When you establish, for example, a 95% confidence interval, you're saying that if the same population is sampled multiple times and the interval calculated each time, 95% of those intervals will include the true population mean. Choosing a confidence level is a balance between precision and certainty. In our household income case, the width of the confidence intervals increases with higher confidence levels because we demand more certainty that the interval contains the population mean, accepting in exchange less precision since the range of the interval becomes wider.