Problem
Heights $(\mathrm{cm})$ and weights $(\mathrm{kg})$ are measured for 100 randomly selected adult males, and range from heights of 130 to $188 \mathrm{~cm}$ and weights of 40 to $150 \mathrm{~kg}$. Let the predictor variable $\mathrm{x}$ be the first variable given. The 100 paired measurements yield $\bar{x}=167.25 \mathrm{~cm}, \bar{y}=81.37 \mathrm{~kg}, r=0.405$, P-value $=0.000$, and $\hat{y}=-106+1.02 x$. Find the best predicted value of $\hat{y}$ (weight) given an adult male who is $166 \mathrm{~cm}$ tall. Use a 0.01 significance level. E) Click the icon to view the critical values of the Pearson correlation coefficient $r$. The best predicted value of $\hat{y}$ for an adult male who is $166 \mathrm{~cm}$ tall is $\square \mathrm{kg}$. (Round to two decimal places as needed.)
Solution
Step 1 ：Given the equation for the predicted weight, which is \(\hat{y}=-106+1.02 x\), where \(x\) is the height in cm.
Step 2 ：Substitute \(x=166\) into the equation to find the predicted weight.
Step 3 ：Calculate \(\hat{y}=-106+1.02 \times 166\) to get \(\hat{y}=63.31999999999999\).
Step 4 ：Round \(\hat{y}\) to two decimal places to get \(\hat{y}=63.32\).
Step 5 ：Final Answer: The best predicted value of \(\hat{y}\) for an adult male who is 166 cm tall is \(\boxed{63.32 kg}\).
