Making Predictions
In Exercises 5–8, let the predictor variable x be the first variable given. Use the given data to find the regression equation and the best predicted value of the response variable. Be sure to follow the prediction procedure summarized in Figure 10-5. Use a 0.05 significance level.


Bear Measurements Head widths (in.) and weights (lb) were measured for 20 randomly selected bears (from Data Set 18 “Bear Measurements” in Appendix B). The 20 pairs of measurements yield xbar = 6.9 in., ybar = 214.3 lb, r = 0.879 P-value = 0.000 and y^ = -212 + 61.9x. Find the best predicted weight of a bear given that the bear has a head width of 6.5 in.

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with predicting the weight of a bear (response variable y) given its head width (predictor variable x = 6.5 in.) using the provided regression equation ŷ = -212 + 61.9x. Additionally, we need to ensure the regression model is appropriate for making predictions by following the prediction procedure outlined in Figure 10-5.
2. Step 2: Verify the significance of the regression model. The problem states that the P-value is 0.000, which is less than the significance level of 0.05. This indicates that the regression model is statistically significant and can be used for making predictions.
3. Step 3: Check whether the given x-value (6.5 in.) is within the scope of the data. The mean head width (x̄) is 6.9 in., and since the x-value of 6.5 in. is reasonably close to the mean and likely within the range of the data, it is appropriate to use the regression equation for prediction.
4. Step 4: Substitute the given x-value (6.5 in.) into the regression equation ŷ = -212 + 61.9x. This involves replacing x with 6.5 and simplifying the expression: ŷ = -212 + 61.9(6.5).
5. Step 5: Simplify the expression to calculate the predicted weight ŷ. This will give the best predicted weight of the bear with a head width of 6.5 in. Ensure the units of the prediction are consistent with the response variable (pounds).

Key Concepts

Regression Equation

A regression equation models the relationship between a predictor variable (x) and a response variable (y). In this case, the equation y^ = -212 + 61.9x indicates how changes in head width (x) affect the predicted weight (y) of the bear. The coefficients represent the intercept and slope, respectively, allowing for predictions based on the value of x.

Significance Level

The significance level, often denoted as alpha (α), is a threshold used to determine whether the results of a statistical test are significant. In this context, a 0.05 significance level means that there is a 5% risk of concluding that a relationship exists when there is none. It helps in making decisions about the validity of the regression model based on the p-value.

Best Predicted Value

The best predicted value refers to the estimated response variable (y) obtained by substituting a specific value of the predictor variable (x) into the regression equation. For example, to find the predicted weight of a bear with a head width of 6.5 inches, you would substitute x = 6.5 into the regression equation, yielding the best estimate of the bear's weight based on the model.