In a random sample of 12 senior-level civil engineers, the mean annual earnings were \$133,326 and the standard deviation was \$36,729. Assume the annual earnings are normally distributed and construct a 95% confidence interval for the population mean annual earnings for senior-level civil engineers. Interpret the results. (Adapted from Salary.com)

Verified step by step guidance

1. Step 1: Identify the given values. From the problem, the sample mean (\( \bar{x} \)) is \$133,326, the sample standard deviation (\( s \)) is \$36,729, the sample size (\( n \)) is 12, and the confidence level is 95%.
2. Step 2: Determine the critical value (\( t^* \)) for a 95% confidence interval. Since the sample size is small (n < 30) and the population standard deviation is unknown, use the t-distribution. The degrees of freedom (df) are \( n - 1 = 12 - 1 = 11 \). Look up the critical value \( t^* \) in a t-distribution table or use statistical software for a two-tailed test with 95% confidence and 11 degrees of freedom.
3. Step 3: Calculate the standard error of the mean (SE). The formula for the standard error is \( SE = \frac{s}{\sqrt{n}} \), where \( s \) is the sample standard deviation and \( n \) is the sample size.
4. Step 4: Compute the margin of error (ME). The formula for the margin of error is \( ME = t^* \cdot SE \), where \( t^* \) is the critical value from Step 2 and \( SE \) is the standard error from Step 3.
5. Step 5: Construct the confidence interval. The formula for the confidence interval is \( \bar{x} \pm ME \), where \( \bar{x} \) is the sample mean and \( ME \) is the margin of error. Interpret the results by stating that you are 95% confident the true population mean annual earnings for senior-level civil engineers fall within this interval.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. In this case, a 95% confidence interval means that if we were to take many samples and construct intervals in the same way, approximately 95% of those intervals would contain the true mean annual earnings of senior-level civil engineers.

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. In this scenario, assuming that the annual earnings are normally distributed allows us to use the properties of the normal distribution to calculate the confidence interval for the population mean.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. In this question, the standard deviation of $36,729 provides insight into the variability of annual earnings among senior-level civil engineers.