Question

police sometimes measure shoe prints at crime scenes so that they can learn something about criminals. listed belo are shoe print lengths, foot lengths, and heights of males. construct a scatterplot, find the value of the linear correlatio 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 = 0.843.
(round to three decimal places as needed.)
determine the null and alternative hypotheses.
h₀: ρ = 0
h₁: ρ ≠ 0
(type integers or decimals. do not round.)
the test statistic is t = □.
(round to two decimal places as needed.)

Explanation:

Step1: Recall the formula for t - statistic in correlation

The formula for the test statistic \( t \) when testing the significance of the linear correlation coefficient \( r \) is \( t=\frac{r\sqrt{n - 2}}{\sqrt{1-r^{2}}} \), where \( n \) is the number of pairs of data. Here, \( r = 0.843 \) and \( n=5 \) (since there are 5 pairs of shoe print length and height data).

Step2: Substitute the values into the formula

First, calculate \( n - 2=5 - 2 = 3 \). Then, calculate the numerator: \( r\sqrt{n - 2}=0.843\times\sqrt{3}\approx0.843\times1.732\approx1.460 \). Next, calculate the denominator: \( \sqrt{1 - r^{2}}=\sqrt{1-(0.843)^{2}}=\sqrt{1 - 0.710649}=\sqrt{0.289351}\approx0.538 \). Then, \( t=\frac{1.460}{0.538}\approx2.71 \) (rounded to two decimal places).

Answer:

\( 2.71 \)