Credit Scores Use the results of Problem 12 from Section 12.3 to answer the following questions:
d. Construct a 90% prediction interval for the interest rate of Kaleigh, whose credit score is 730.

Verified step by step guidance

1. Identify the regression equation from Problem 12 in Section 12.3, which relates the interest rate (dependent variable) to the credit score (independent variable). The equation will be of the form \(\hat{y} = b_0 + b_1 x\), where \(b_0\) is the intercept and \(b_1\) is the slope.
2. Calculate the predicted interest rate \(\hat{y}\) for Kaleigh's credit score of 730 by substituting \(x = 730\) into the regression equation: \(\hat{y} = b_0 + b_1 \times 730\).
3. Find 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: \(SE_{pred} = s \sqrt{1 + \frac{1}{n} + \frac{(x_0 - \bar{x})^2}{\sum (x_i - \bar{x})^2}}\) where \(s\) is the standard error of the regression, \(n\) is the sample size, \(x_0\) is 730, and \(\bar{x}\) is the mean of the credit scores in the sample.
4. Determine the critical value from the t-distribution for a 90% prediction interval with \(n - 2\) degrees of freedom. This value is denoted as \(t^*\) and can be found using statistical tables or software.
5. Construct the 90% prediction interval using the formula: \(\hat{y} \pm t^* \times SE_{pred}\) This interval gives the range in which we expect Kaleigh's actual interest rate to fall with 90% 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 outcomes.

Linear Regression and Regression Equation

Linear regression models the relationship between a dependent variable and one or more independent variables using a straight line. The regression equation predicts the expected value of the dependent variable based on given independent variable values, such as predicting interest rate from credit score.

Confidence Level and Its Role in Interval Estimation

The confidence level (e.g., 90%) represents the proportion of similarly constructed intervals that would contain the true value if the experiment were repeated many times. It reflects the degree of certainty in the interval estimate and influences the width of the prediction interval.