Answer :
To determine which statement is true, we can use the given equation for the least-squares line: [tex]\(\hat{y} = 0.1 + 3.1x\)[/tex]. In this equation, [tex]\(\hat{y}\)[/tex] represents the predicted circumference, and [tex]\(x\)[/tex] is the tire diameter at the tread. We will calculate the circumference for each diameter given in the statements and check if it matches the statement's claim:
1. Statement a: A tire with a 30-inch diameter at the tread will have a circumference of 93.1 inches.
Substitute [tex]\(x = 30\)[/tex] into the equation:
[tex]\[
\hat{y} = 0.1 + 3.1 \times 30 = 0.1 + 93 = 93.1
\][/tex]
The calculated circumference is 93.1 inches, which matches the statement. Thus, statement a is true.
2. Statement b: A tire with a 32-inch diameter at the tread will have a circumference of 86.9 inches.
Substitute [tex]\(x = 32\)[/tex] into the equation:
[tex]\[
\hat{y} = 0.1 + 3.1 \times 32 = 0.1 + 99.2 = 99.3
\][/tex]
The calculated circumference is 99.3 inches, which does not match the statement. Therefore, statement b is false.
3. Statement c: A tire with a 35-inch diameter at the tread will have a circumference of 124.1 inches.
Substitute [tex]\(x = 35\)[/tex] into the equation:
[tex]\[
\hat{y} = 0.1 + 3.1 \times 35 = 0.1 + 108.5 = 108.6
\][/tex]
The calculated circumference is 108.6 inches, which does not match the statement. Therefore, statement c is false.
4. Statement d: A tire with a 28-inch diameter at the tread will have a circumference of 99.3 inches.
Substitute [tex]\(x = 28\)[/tex] into the equation:
[tex]\[
\hat{y} = 0.1 + 3.1 \times 28 = 0.1 + 86.8 = 86.9
\][/tex]
The calculated circumference is 86.9 inches, which does not match the statement. Therefore, statement d is false.
The only true statement is a, where a tire with a 30-inch diameter has a circumference of 93.1 inches.
1. Statement a: A tire with a 30-inch diameter at the tread will have a circumference of 93.1 inches.
Substitute [tex]\(x = 30\)[/tex] into the equation:
[tex]\[
\hat{y} = 0.1 + 3.1 \times 30 = 0.1 + 93 = 93.1
\][/tex]
The calculated circumference is 93.1 inches, which matches the statement. Thus, statement a is true.
2. Statement b: A tire with a 32-inch diameter at the tread will have a circumference of 86.9 inches.
Substitute [tex]\(x = 32\)[/tex] into the equation:
[tex]\[
\hat{y} = 0.1 + 3.1 \times 32 = 0.1 + 99.2 = 99.3
\][/tex]
The calculated circumference is 99.3 inches, which does not match the statement. Therefore, statement b is false.
3. Statement c: A tire with a 35-inch diameter at the tread will have a circumference of 124.1 inches.
Substitute [tex]\(x = 35\)[/tex] into the equation:
[tex]\[
\hat{y} = 0.1 + 3.1 \times 35 = 0.1 + 108.5 = 108.6
\][/tex]
The calculated circumference is 108.6 inches, which does not match the statement. Therefore, statement c is false.
4. Statement d: A tire with a 28-inch diameter at the tread will have a circumference of 99.3 inches.
Substitute [tex]\(x = 28\)[/tex] into the equation:
[tex]\[
\hat{y} = 0.1 + 3.1 \times 28 = 0.1 + 86.8 = 86.9
\][/tex]
The calculated circumference is 86.9 inches, which does not match the statement. Therefore, statement d is false.
The only true statement is a, where a tire with a 30-inch diameter has a circumference of 93.1 inches.
