American Black Bears Use the results of Problem 18 from Section 12.3 to answer the following questions:
d. Construct a 95% prediction interval for the weight of an American black bear that is 154.5 cm long.

Verified step by step guidance

1. Identify the regression equation from Problem 18 in Section 12.3, which relates the length of an American black bear to its weight. This equation typically has the form \(\hat{y} = b_0 + b_1 x\), where \(\hat{y}\) is the predicted weight and \(x\) is the length (154.5 cm in this case).
2. Calculate the predicted weight \(\hat{y}\) by substituting \(x = 154.5\) cm into the regression equation: \(\hat{y} = b_0 + b_1 \times 154.5\).
3. Determine the standard error of the prediction, which accounts for both the variability of the regression line and the individual prediction. The formula for the standard error of prediction is: \(SE_{pred} = s \sqrt{1 + \frac{1}{n} + \frac{(x - \bar{x})^2}{\sum (x_i - \bar{x})^2}}\) where: - \(s\) is the standard error of the estimate, - \(n\) is the sample size, - \(\bar{x}\) is the mean of the \(x\) values, - \(x_i\) are the observed \(x\) values.
4. Find the critical value \(t^*\) from the \(t\)-distribution with \(n-2\) degrees of freedom corresponding to a 95% confidence level. This value accounts for the desired confidence in the prediction interval.
5. Construct the 95% prediction interval using the formula: \(\hat{y} \pm t^* \times SE_{pred}\) This interval gives a range where we expect the weight of a bear with length 154.5 cm to fall with 95% confidence.

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 (e.g., 95%). Unlike confidence intervals for the mean, prediction intervals account for both the uncertainty in estimating the mean and the variability of individual data points.

Linear Regression Model

A linear regression model describes the relationship between a dependent variable (e.g., bear weight) and an independent variable (e.g., bear length) using a linear equation. It allows prediction of the dependent variable based on new values of the independent variable.

Confidence Level and t-Distribution

The confidence level (e.g., 95%) indicates the probability that the interval contains the true value. When sample sizes are small or population variance is unknown, the t-distribution is used instead of the normal distribution to calculate critical values for intervals.