Heights (cm) and weights (kg) are measured for 100 randomly selected adult males, with heights ranging from 139 to 192 cm and weights from 40 to 150 kg. Let the predictor variable be [tex]$ x $[/tex] with [tex]$ r = 0.314 $[/tex], P-value [tex]$ = 0.001 $[/tex], and [tex]$\hat{y} = -101 + 1.19x$[/tex].

Find the best predicted value of [tex]$\hat{y}$[/tex] (weight) given an adult male who is 187 cm tall. Use a 0.05 significance level.

The best predicted value of [tex]$\hat{y}$[/tex] for an adult male who is 187 cm tall is [tex]$\square$[/tex] kg. (Round to two decimal places as needed.)

To find the best predicted value of the weight ([tex]$\hat{y}$[/tex]) for an adult male who is 187 cm tall, we use the linear regression equation [tex]$\hat{y} = -101 + 1.19x$[/tex], where [tex]$x$[/tex] is the height in centimeters.

Here's how to solve the problem step-by-step:

1. Identify the Variables:
- Predictor variable ([tex]$x$[/tex]): Height of the adult male, which is 187 cm.
- Intercept of the regression line: [tex]$-101$[/tex].
- Slope of the regression line: [tex]$1.19$[/tex].

2. Use the Regression Equation:
The linear regression equation is [tex]$\hat{y} = -101 + 1.19x$[/tex]. Our goal is to plug in the given height into this equation to find the predicted weight.

3. Perform the Calculation:
Substitute the height ([tex]$x = 187$[/tex]) into the equation:
[tex]\[
\hat{y} = -101 + 1.19 \times 187
\][/tex]

4. Calculate Step-by-Step:
- First, multiply the slope ([tex]$1.19$[/tex]) by the height ([tex]$187$[/tex]):
[tex]\[
1.19 \times 187 = 222.53
\][/tex]
- Next, add the intercept ([tex]$-101$[/tex]) to this product:
[tex]\[
\hat{y} = -101 + 222.53 = 121.53
\][/tex]

5. Round the Result:
The predicted value [tex]$\hat{y}$[/tex] is already rounded to two decimal places, which gives us:
[tex]\[
\hat{y} = 121.53 \text{ kg}
\][/tex]

Therefore, the best predicted value of the weight for an adult male who is 187 cm tall is 121.53 kg.