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Answer :
To find the best predicted value of [tex]\(\hat{y}\)[/tex] (weight) for an adult male who is 182 cm tall, we use the provided linear regression equation:
[tex]\[
\hat{y} = -109 + 1.07x
\][/tex]
Here, [tex]\(\hat{y}\)[/tex] represents the predicted weight, and [tex]\(x\)[/tex] represents the height in centimeters.
Step-by-step Solution:
1. Identify the given height: The problem states that we need to predict the weight for an adult male who is 182 cm tall. Thus, [tex]\(x = 182\)[/tex].
2. Substitute the height into the regression equation: Replace [tex]\(x\)[/tex] in the equation with 182:
[tex]\[
\hat{y} = -109 + 1.07 \times 182
\][/tex]
3. Calculate the predicted weight: Perform the multiplication and addition:
- First, calculate [tex]\(1.07 \times 182\)[/tex]:
[tex]\[
1.07 \times 182 = 194.74
\][/tex]
- Then add [tex]\(-109\)[/tex] to this result:
[tex]\[
\hat{y} = -109 + 194.74 = 85.74
\][/tex]
Thus, the best predicted value of [tex]\(\hat{y}\)[/tex] for an adult male who is 182 cm tall is [tex]\(85.74\)[/tex] kg, rounded to two decimal places.
[tex]\[
\hat{y} = -109 + 1.07x
\][/tex]
Here, [tex]\(\hat{y}\)[/tex] represents the predicted weight, and [tex]\(x\)[/tex] represents the height in centimeters.
Step-by-step Solution:
1. Identify the given height: The problem states that we need to predict the weight for an adult male who is 182 cm tall. Thus, [tex]\(x = 182\)[/tex].
2. Substitute the height into the regression equation: Replace [tex]\(x\)[/tex] in the equation with 182:
[tex]\[
\hat{y} = -109 + 1.07 \times 182
\][/tex]
3. Calculate the predicted weight: Perform the multiplication and addition:
- First, calculate [tex]\(1.07 \times 182\)[/tex]:
[tex]\[
1.07 \times 182 = 194.74
\][/tex]
- Then add [tex]\(-109\)[/tex] to this result:
[tex]\[
\hat{y} = -109 + 194.74 = 85.74
\][/tex]
Thus, the best predicted value of [tex]\(\hat{y}\)[/tex] for an adult male who is 182 cm tall is [tex]\(85.74\)[/tex] kg, rounded to two decimal places.
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