Question

find the equation of the regression line for the given data. then construct a scatter plot of the data and draw the regression line. (the pair of variables has a significant correlation.) then use the regression equation to predict the value of \\(y\\) for each of the given \\(x\\)-values, if meaningful. the table shows the shoe size and heights (in) for 6 men.

\\(\

$$\begin{array}{|l|c|c|c|c|c|c|}\\hline \\text{shoe size, } x & 8.5 & 9.5 & 10.0 & 12.0 & 12.5 & 13.5 \\\\ \\hline \\text{height, } y & 65.0 & 67.0 & 72.0 & 72.0 & 73.0 & 72.0 \\\\ \\hline \\end{array}$$

\\)

(a) \\(x = \text{size } 11.5\\)
(b) \\(x = \text{size } 10.5\\)
(c) \\(x = \text{size } 16.5\\)
(d) \\(x = \text{size } 9.0\\)

(a) predict the value of \\(y\\) for \\(x = 11.5\\). choose the correct answer below.

a. 69.5
b. 67.4
c. 70.9
d. not meaningful

(b) predict the value of \\(y\\) for \\(x = 10.5\\). choose the correct answer below.

a. 69.5
b. 77.8
c. 67.4
d. not meaningful

Explanation:

⚡ Using what you learned: regression, residuals, and model fit

Step 1: Calculate Summary Statistics

Find the sample size \( n \), and the sums of \( x \), \( y \), \( x^2 \), \( y^2 \), and \( xy \).

The given data points \( (x, y) \) are:

$$ (8.5, 65.0), (9.5, 67.0), (10.0, 72.0), (12.0, 72.0), (12.5, 73.0), (13.5, 72.0) $$

Number of data points:

$$ n = 6 $$

Calculate the sums:

$$ \sum x = 8.5 + 9.5 + 10.0 + 12.0 + 12.5 + 13.5 = 66.0 $$

$$ \sum y = 65.0 + 67.0 + 72.0 + 72.0 + 73.0 + 72.0 = 421.0 $$

$$ \sum x^2 = 8.5^2 + 9.5^2 + 10.0^2 + 12.0^2 + 12.5^2 + 13.5^2 = 72.25 + 90.25 + 100.0 + 144.0 + 156.25 + 182.25 = 745.0 $$

$$ \sum xy = (8.5 \times 65.0) + (9.5 \times 67.0) + (10.0 \times 72.0) + (12.0 \times 72.0) + (12.5 \times 73.0) + (13.5 \times 72.0) $$

$$ \sum xy = 552.5 + 636.5 + 720.0 + 864.0 + 912.5 + 972.0 = 4657.5 $$

Step 2: Find the Regression Equation

Calculate the slope \( m \) and \( y \)-intercept \( b \) for the line \( \hat{y} = mx + b \):

$$ m = \frac{n \sum xy - (\sum x)(\sum y)}{n \sum x^2 - (\sum x)^2} $$

$$ m = \frac{6(4657.5) - (66.0)(421.0)}{6(745.0) - (66.0)^2} $$

$$ m = \frac{27945 - 27786}{4470 - 4356} = \frac{159}{114} \approx 1.3947 $$

$$ b = \frac{\sum y - m \sum x}{n} $$

$$ b = \frac{421.0 - (1.3947)(66.0)}{6} \approx \frac{421.0 - 92.0502}{6} \approx \frac{328.9498}{6} \approx 54.825 $$

Thus, the regression equation is:

$$ \hat{y} = 1.395x + 54.825 $$

Step 3: Predict the value of \( y \) for \( x = 11.5 \)

Substitute \( x = 11.5 \) into the regression equation:

$$ \hat{y} = 1.3947(11.5) + 54.825 \approx 16.039 + 54.825 = 70.864 \approx 70.9 $$

This matches option C.

Step 4: Predict the value of \( y \) for \( x = 10.5 \)

Substitute \( x = 10.5 \) into the regression equation:

$$ \hat{y} = 1.3947(10.5) + 54.825 \approx 14.644 + 54.825 = 69.469 \approx 69.5 $$

This matches option A.

Answer:

(a) C. 70.9
(b) A. 69.5