"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.
22. Mean Hourly Wage Construct a 95% prediction interval for the mean hourly wage in Exercise 12 when the median hourly wage is \$21.50."

Verified step by step guidance

1. Identify the key components needed to construct a prediction interval: the sample mean (\( \bar{x} \)), the sample standard deviation (\( s \)), the sample size (\( n \)), and the desired confidence level (95% in this case).
2. Recall that a prediction interval estimates the range in which a single new observation is likely to fall, not the mean of the population. The formula for a prediction interval is: \( \bar{x} \pm t_{\alpha/2, n-1} \times s \times \sqrt{1 + \frac{1}{n}} \), where \( t_{\alpha/2, n-1} \) is the critical value from the t-distribution with \( n-1 \) degrees of freedom.
3. Find the critical t-value (\( t_{\alpha/2, n-1} \)) corresponding to a 95% confidence level and \( n-1 \) degrees of freedom using a t-table or statistical software.
4. Plug in the values for \( \bar{x} \), \( s \), \( n \), and the critical t-value into the prediction interval formula to calculate the lower and upper bounds of the interval.
5. Interpret the prediction interval by explaining that we are 95% confident that the hourly wage of a single new worker will fall within this interval, considering the variability in the data.

Key Concepts

Prediction Interval

A prediction interval estimates the range within which a single new observation is expected to fall, with a certain level of confidence. Unlike confidence intervals for means, prediction intervals account for both the variability in the sample mean and the individual data points, making them wider and more appropriate for predicting individual outcomes.

Confidence Level (e.g., 95%)

The confidence level represents the probability that the interval constructed from sample data contains the true parameter or future observation. A 95% confidence level means that if we repeated the sampling many times, about 95% of the prediction intervals would include the actual value.

Relationship Between Median and Mean in Wage Data

Understanding the difference between median and mean wages is important because the mean can be influenced by extreme values, while the median represents the middle value. When constructing prediction intervals for the mean hourly wage, knowing the median helps contextualize the distribution but does not replace the mean used in calculations.