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\\)

(c) predict the value of \\(y\\) for \\(x = 16.5\\). choose the correct answer below.

a. 77.8
b. 69.5
c. 70.9
d. not meaningful

(d) predict the value of \\(y\\) for \\(x = 9.0\\). choose the correct answer below.

a. 70.9
b. 77.8
c. 67.4
d. not meaningful

Explanation:

⚡ Using what you learned: regression, residuals, and model fit · 🆕 New Concept Discovered: Extrapolation in Regression
Using regression equations only within the range of observed data.

Step 1: Find the regression equation

First, we calculate the summary statistics for the given data pairs \((x, y)\):

• Shoe size, \(x\): \(8.5, 9.5, 10.0, 12.0, 12.5, 13.5\)
• Height, \(y\): \(65.0, 67.0, 72.0, 72.0, 73.0, 72.0\)

Number of data points \(n = 6\).

Calculate the sums:

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

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

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

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

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

Calculate the slope \(m\):

$$ m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2} = \frac{6(4657.5) - (66.0)(421.0)}{6(745.0) - (66.0)^2} $$

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

Calculate the y-intercept \(b\):

$$ b = \frac{\sum y - m \sum x}{n} = \frac{421.0 - 1.3947(66.0)}{6} \approx \frac{421.0 - 92.05}{6} \approx 54.825 $$

The regression equation is:

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

Step 2: Evaluate prediction for \(x = 16.5\) (part c)

The range of the observed independent variable \(x\) (shoe size) is from \(8.5\) to \(13.5\).

Predicting a value for \(x = 16.5\) is far outside the range of the sample data. In statistics, making predictions outside the range of observed \(x\)-values is called extrapolation and is generally considered unreliable or not meaningful because we cannot assume the linear relationship continues to hold.

Therefore, the prediction for \(x = 16.5\) is not meaningful.

Step 3: Evaluate prediction for \(x = 9.0\) (part d)

The value \(x = 9.0\) lies within the range of our observed data \([8.5, 13.5]\), so making a prediction is meaningful.

Substitute \(x = 9.0\) into the regression equation:

$$ \hat{y} = 1.3947(9.0) + 54.825 $$

$$ \hat{y} = 12.5523 + 54.825 \approx 67.377 $$

Rounding to one decimal place gives \(67.4\).

Answer:

(c) D. not meaningful
(d) C. 67.4