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 on page 493. Use a 0.05 significance level.

Head widths (in.) and weights (lb) were measured for 20 randomly selected bears (from Data Set 9 “Bear Measurements” in Appendix B). The 20 pairs of measurements yield\(\bar x = 6.9\)in.,\(\bar y = 214.3\)lb, r= 0.879, P-value = 0.000, and\(\hat y = - 212 + 61.9x\). Find the best predicted value of\(\hat y\)(weight) given a bear with a head width of 6.5 in.

The sample number of bears is\(n = 20\). x represents thehead widths and y represents head weights.

The mean head width and weight are \(\bar x = 6.9\) and \(\bar y = 214.3\). The correlation coefficient is \(r = 0.879\) and the P-value is 0.000. The regression equation is \(\hat y = - 212 + 61.9x\).

The statistical hypotheses are formed as,

\({H_0}:\)The correlation coefficient is not significant.

\({H_1}:\)The correlation coefficient is significant.

Since the P-value (0.000) is less than the level of significance (0.05). In this case, the null hypothesis is rejected.

Therefore,the correlation coefficient is significant.

Referring to Figure 10-5, the regression model is a good model and thus the regression equation can be used to predict the value of y.

Thepredicted value is computed as,

\(\begin{array}{c}\hat y = - 212 + \left( {61.9 \times 6.5} \right)\\ = - 212 + 402.35\\ = 190.35\end{array}\).

Thus, the predicted value of the \(\hat y\)(weight) for a bear with a head width of 6.5 in is 190.35 lb.