AARP reported on a study conducted to learn how long it takes individuals to prepare their federal income tax return \((A A R P \text { Bulletin, April } 2008\) ). The data contained in the file named TaxReturn are consistent with the study results. These data provide the time in hours required for 40 individuals to complete their federal income tax returns. Using past years' data, the population standard deviation can be assumed known with \(\sigma=9\) hours. What is the \(95 \%\) confidence interval estimate of the mean time it takes an individual to complete a federal income tax return?

When you encounter the term population standard deviation, it refers to the measure of variation or spread in a set of values. In simpler terms, it quantifies how much the numbers in a complete population differ from the population's average value, known as the mean.

In our exercise, the population standard deviation is denoted by the Greek letter sigma \( \sigma \) and is given as 9 hours. This means, on average, the individual completion times vary by 9 hours from the mean completion time for the entire population. Knowing this value is crucial because it's used to calculate the standard error when constructing confidence intervals—if the standard deviation is large, the spread of individual times is more considerable, which would typically lead to a broader confidence interval.

The sample mean calculation is a fundamental step in statistical analysis. It represents the average of a sample and is computed by summing all the measurements in the sample and then dividing by the number of observations.

In the provided exercise, the sample mean, represented as \( \bar{x} \), is the average time it takes for the 40 individuals to complete their tax returns. Although the exact data points aren't listed, we're informed that the sample mean \( \bar{x} \) was obtained by summing the tax return completion times for all individuals involved and then dividing by the total number of individuals, which is 40 in this case. This value is a critical component in finding our confidence interval estimate.

A Z-score is a statistical metric that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations. It is used in the construction of confidence intervals to determine how far, and in what direction, the data deviates from the mean.

For a 95% confidence interval, the Z-score corresponds to the point on the normal distribution that captures 95% of all possible sample means. In the exercise, the Z-score of approximately 1.96 tells us that 95% of all possible sample means fall within 1.96 standard deviations of the population mean. This specific Z-score is a critical value for calculating the margin of error.

The standard error (SE) quantifies the variability of the sample mean estimate of a population parameter. It essentially tells us how far the sample mean is likely to be from the true population mean.

To calculate it, we divide the population standard deviation by the square root of the number of observations in the sample \( \sqrt{n} \). In the given exercise, the standard error helps in assessing the extent to which the sample mean (\( \bar{x} \) ) may differ from the actual population mean. It is a factor in determining the width of the confidence interval: the larger the standard error, the less certain we are about the sample mean representing the true population mean, thus leading to a wider interval.

Lastly, the margin of error (ME) reflects the range above and below the sample mean within which we can expect the actual population mean to lie. It accounts for possible sampling variability and is influenced by the standard error and the confidence level of the interval.

In the step-by-step solution, we multiply the Z-score by the standard error to find the margin of error. This margin determines how much room there is for potential discrepancies between the sample mean and the true population mean. The calculated margin of error then helps us create the confidence interval by adding and subtracting it from the sample mean, giving us a range that should contain the true mean with 95% certainty.