"In Exercises 19-24, construct the indicated prediction interval and interpret the results.
19. Construct a 90% prediction interval for the amount of milk produced in Exercise 9 when there are an average of 9275 thousand milk cows."

Verified step by step guidance

1. Identify the regression model from Exercise 9, which relates the number of milk cows to the amount of milk produced. This model typically has the form: $\mathrm{\backslash hat\{y\}}=b+mx$, where $\mathrm{\backslash hat\{y\}}$ is the predicted milk production and $x$ is the number of milk cows (in thousands).
2. Calculate the predicted value of milk production, $\mathrm{\backslash hat\{y\}}$, by substituting $x=\; 9275$ 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 is: $\mathrm{SE\_\{pred\}\; =\; s\; \backslash sqrt\{1\; +\; \backslash frac\{1\}\{n\}\; +\; \backslash frac\{(x\_0\; -\; \backslash bar\{x\})^2\}\{\backslash sum\; (x\_i\; -\; \backslash bar\{x\})^2\}\}}$, where $s$ is the standard error of the estimate, $n$ is the sample size, $x\_0$ is 9275, and $\backslash bar\{x\}$ is the mean of the $x$ values.
4. Find the critical t-value for a 90% prediction interval with degrees of freedom equal to $n\; -\; 2$. This value comes from the t-distribution table and corresponds to the desired confidence level.
5. Construct the 90% prediction interval using the formula: $\mathrm{\backslash hat\{y\}}\backslash pm\; t\_\{\backslash alpha/2,\; n-2\}\; \backslash times\; SE\_\{pred\}$. This interval estimates the range in which a single new observation of milk production is expected to fall when there are 9275 thousand milk cows.

Key Concepts

Prediction Interval

A prediction interval estimates the range within which a single future 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 observations.

Confidence Level

The confidence level, such as 90%, represents the probability that the prediction interval contains the true future value. It reflects the degree of certainty we have in the interval estimate, with higher levels indicating wider intervals to capture more uncertainty.

Regression and Use of Predictor Variables

Prediction intervals often rely on regression models that relate a predictor variable (e.g., number of milk cows) to a response variable (e.g., milk production). Understanding how to use the regression equation and the variability around the predicted value is essential to construct accurate prediction intervals.