Constructing Confidence Intervals In Exercises 13–24, assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance σ^2. Interpret the results.
[APPLET] Earnings The annual earnings (in thousands of dollars) of 21 randomly selected level 1 computer hardware engineers are listed. Use a 99% level of confidence. (Adapted from Salary.com)

Verified step by step guidance

1. Step 1: Calculate the sample variance (s^2). First, compute the sample mean (x̄) by summing all the earnings values and dividing by the sample size (n = 21). Then, use the formula for variance: s^2 = (Σ(xi - x̄)^2) / (n - 1), where xi represents each individual data point.
2. Step 2: Identify the degrees of freedom (df). For this problem, df = n - 1, where n is the sample size. Since n = 21, df = 20.
3. Step 3: Use the chi-square distribution to find the critical values for the 99% confidence interval. The chi-square critical values are determined using the chi-square table or a statistical calculator for df = 20 and the desired confidence level (99%). The lower critical value corresponds to α/2, and the upper critical value corresponds to 1 - α/2, where α = 0.01.
4. Step 4: Construct the confidence interval for the population variance (σ^2) using the formula: [(df * s^2) / χ²_upper, (df * s^2) / χ²_lower], where χ²_upper and χ²_lower are the chi-square critical values.
5. Step 5: Interpret the results. The confidence interval provides a range of values within which the true population variance is likely to fall with 99% confidence. This interval reflects the variability in annual earnings for level 1 computer hardware engineers.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence. For example, a 99% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 99% of those intervals would contain the true population parameter.

Population Variance

Population variance is a measure of the dispersion of a set of values in a population, indicating how much the values deviate from the population mean. It is denoted by σ² and is calculated as the average of the squared differences from the mean. Understanding population variance is crucial for constructing confidence intervals for variance.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. Many statistical methods, including confidence interval calculations, assume that the underlying data follows a normal distribution, especially when sample sizes are small.