"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.
25. Mean Wage Construct a 99% prediction interval for the mean annual wage in Exercise 15 when the percentage of employment in STEM occupations is 13% in the industry."

Verified step by step guidance

1. Identify the regression equation from Exercise 15, which relates the mean annual wage (dependent variable) to the percentage of employment in STEM occupations (independent variable). This equation typically has the form: $\mathrm{\backslash (\backslash hat\{y\}\backslash )}=b\_0\; +b\_1x$, where $\mathrm{\backslash (\backslash hat\{y\}\backslash )}$ is the predicted mean wage, $b\_0$ is the intercept, $b\_1$ is the slope, and $x$ is the percentage of employment in STEM occupations.
2. Calculate the predicted mean wage $\mathrm{\backslash (\backslash hat\{y\}\backslash )}$ for $x=\; 13\%$ by substituting 13 into the regression equation.
3. Determine the standard error of the prediction, which accounts for both the variability of the estimate of the mean response and the variability of individual observations. The formula for the standard error of prediction at $x\; =\; x\_0$ is: $SE\_\{pred\}\; =\; s\; \backslash (\backslash sqrt\{1\; +\; \backslash frac\{1\}\{n\}\backslash )\; +\; \backslash (\backslash frac\{(x\_0\; -\; \backslash bar\{x\}\backslash ))^2\}\{\backslash (\backslash sum\backslash )\; (x\_i\; -\; \backslash (\backslash bar\{x\}\backslash ))^2\}\}$, where $s$ is the standard error of the estimate, $n$ is the sample size, $\backslash (\backslash bar\{x\}\backslash )$ is the mean of the $x$ values, and $x\_0\; =\; 13$.
4. Find the critical t-value for a 99% prediction interval with $n\; -\; 2$ degrees of freedom from the t-distribution table. This value corresponds to the desired confidence level and sample size.
5. Construct the 99% prediction interval using the formula: $\backslash (\backslash hat\{y\}\backslash )\; \backslash (\backslash pm\backslash )\; t\_\{\backslash (\backslash alpha\backslash )/2,\; n-2\}\; \backslash (\backslash times\backslash )\; SE\_\{pred\}$. Interpret this interval as the range in which we expect the annual wage for an individual industry with 13% employment in STEM occupations to fall with 99% confidence.

Key Concepts

Prediction Interval

A prediction interval estimates the range within which a single new observation is expected to fall, with a specified level of confidence. Unlike confidence intervals for the mean, prediction intervals account for both the uncertainty in estimating the mean and the variability of individual data points.

Regression and Predictor Variables

Regression analysis models the relationship between a dependent variable and one or more independent variables. Here, the percentage of employment in STEM occupations is the predictor used to estimate the mean annual wage, allowing us to predict wages based on STEM employment levels.

Confidence Level and Interpretation

The confidence level (e.g., 99%) indicates the probability that the constructed interval contains the true value. Interpreting a 99% prediction interval means we are 99% confident that the wage for a new observation with 13% STEM employment falls within the calculated range.