Question

police sometimes measure shoe prints at crime scenes so that they can learn something about criminals. listed below are shoe print lengths, foot lengths, and heights of males. construct a scatterplot, find the value of the linear correlation coefficient r, and find the p - value of r. determine whether there is sufficient evidence to support a claim of linear correlation between the two variables. based on these results, does it appear that police can use a shoe print length to estimate the height of a male? use a significance level of α = 0.01.
shoe print (cm) | 30.4 30.4 31.1 31.5 27.2
foot length (cm) | 25.2 25.4 27.5 26.3 26.1
height (cm) | 172.6 181.5 182.3 181.7 168.6
the linear correlation coefficient is r =
(round to three decimal places as needed.)

Explanation:

Step1: Identify Variables

Let \( x \) be Shoe Print (cm) with values \( [30.4, 30.4, 31.1, 31.5, 27.2] \) and \( y \) be Height (cm) with values \( [172.6, 181.5, 182.3, 181.7, 168.6] \).

Step2: Calculate Mean of \( x \) and \( y \)

Mean of \( x \): \( \bar{x} = \frac{30.4 + 30.4 + 31.1 + 31.5 + 27.2}{5} = \frac{150.6}{5} = 30.12 \)
Mean of \( y \): \( \bar{y} = \frac{172.6 + 181.5 + 182.3 + 181.7 + 168.6}{5} = \frac{886.7}{5} = 177.34 \)

Step3: Calculate Deviations and Products

For each pair \((x_i, y_i)\), compute \( (x_i - \bar{x}) \), \( (y_i - \bar{y}) \), and their product, also \( (x_i - \bar{x})^2 \) and \( (y_i - \bar{y})^2 \):

• \( (30.4 - 30.12) = 0.28 \), \( (172.6 - 177.34) = -4.74 \), Product: \( 0.28 \times (-4.74) = -1.3272 \), \( (0.28)^2 = 0.0784 \), \( (-4.74)^2 = 22.4676 \)
• \( (30.4 - 30.12) = 0.28 \), \( (181.5 - 177.34) = 4.16 \), Product: \( 0.28 \times 4.16 = 1.1648 \), \( (0.28)^2 = 0.0784 \), \( (4.16)^2 = 17.3056 \)
• \( (31.1 - 30.12) = 0.98 \), \( (182.3 - 177.34) = 4.96 \), Product: \( 0.98 \times 4.96 = 4.8608 \), \( (0.98)^2 = 0.9604 \), \( (4.96)^2 = 24.6016 \)
• \( (31.5 - 30.12) = 1.38 \), \( (181.7 - 177.34) = 4.36 \), Product: \( 1.38 \times 4.36 = 6.0168 \), \( (1.38)^2 = 1.9044 \), \( (4.36)^2 = 19.0096 \)
• \( (27.2 - 30.12) = -2.92 \), \( (168.6 - 177.34) = -8.74 \), Product: \( (-2.92) \times (-8.74) = 25.5208 \), \( (-2.92)^2 = 8.5264 \), \( (-8.74)^2 = 76.3876 \)

Step4: Sum the Products and Squares

Sum of products (\( \sum (x_i - \bar{x})(y_i - \bar{y}) \)):
\( -1.3272 + 1.1648 + 4.8608 + 6.0168 + 25.5208 = 36.236 \)

Sum of \( (x_i - \bar{x})^2 \) (\( \sum (x_i - \bar{x})^2 \)):
\( 0.0784 + 0.0784 + 0.9604 + 1.9044 + 8.5264 = 11.548 \)

Sum of \( (y_i - \bar{y})^2 \) (\( \sum (y_i - \bar{y})^2 \)):
\( 22.4676 + 17.3056 + 24.6016 + 19.0096 + 76.3876 = 159.772 \)

Step5: Calculate Correlation Coefficient \( r \)

Formula: \( r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} \)

Substitute values:
\( r = \frac{36.236}{\sqrt{11.548 \times 159.772}} \)

First, compute denominator: \( \sqrt{11.548 \times 159.772} \approx \sqrt{1845.0} \approx 42.953 \) (more precisely, \( 11.548 \times 159.772 \approx 1845.0 \), square root is \( \approx 42.953 \))

Then, \( r \approx \frac{36.236}{42.953} \approx 0.843 \) (rounded to three decimal places)

Answer:

\( 0.843 \)