Question

Testing for a Linear Correlation. In Exercises 13–28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of A = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.)

CSI Statistics Use the paired foot length and height data from the preceding exercise. Is there sufficient evidence to conclude that there is a linear correlation between foot lengths and heights of males? Based on these results, does it appear that police can use foot length to estimate the height of a male?

Shoe print(cm)	29.7	29.7	31.4	31.8	27.6
Foot length(cm)	25.7	25.4	27.9	26.7	25.1
Height (cm)	175.3	177.8	185.4	175.3	172.7

Short answer

The scatter plot is shown below:

The value of the correlation coefficient is 0.826.

The p-value is 0.085.

There is not enough evidence to support the claim of alinear correlation between foot length and height of males.

As two variables are not linearly associated and the scatter plot does not indicate a clear non-linear trend, the foot length cannot be used to predict the height of males.

Step-by-step solution

Given information

Refer to Exercise 17 for the data onshoe print, foot length, and height of males.

Shoe print(cm)	29.7	29.7	31.4	31.8	27.6
Foot length(cm)	25.7	25.4	27.9	26.7	25.1
Height (cm)	175.3	177.8	185.4	175.3	172.7

Sketch a scatterplot

Paired values are obtainedon a graph with two axes scaled according to the variables.

Steps to sketch a scatterplot:

1. Mark horizontal axis for footlengthand vertical axis for the height of males.
2. Mark each of the points on the graph.
3. The resultant graph is the scatterplot.

Compute the measure of the correlation coefficient

The formula for correlation coefficient is

\(r = \frac{{n\sum {xy} - \left( {\sum x } \right)\left( {\sum y } \right)}}{{\sqrt {n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}} \sqrt {n\left( {\sum {{y^2}} } \right) - {{\left( {\sum y } \right)}^2}} }}\).

Let foot lengthbe defined by variable x and heights of males be defined by variable y.

The valuesare listed in the table below:

x	y	\({x^2}\)	\({y^2}\)	\(xy\)
25.7	175.3	660.49	30730.09	4505.21
25.4	177.8	645.16	31612.84	4516.12
27.9	185.4	778.41	34373.16	5172.66
26.7	175.3	712.89	30730.09	4680.51
25.1	172.7	630.01	29825.29	4334.77
\(\sum x = 130.8\)	\(\sum y = 886.5\)	\(\sum {{x^2}} = 3426.96\)	\(\sum {{y^2} = } \;157271.5\)	\(\sum {xy\; = \;} 23209.27\)

Substitute the values in the formula:

\(\begin{aligned} r &= \frac{{5\left( {23209.27} \right) - \left( {130.8} \right)\left( {886.5} \right)}}{{\sqrt {5\left( {3426.96} \right) - {{\left( {130.8} \right)}^2}} \sqrt {5\left( {157271.5} \right) - {{\left( {886.5} \right)}^2}} }}\\ &= 0.826\end{aligned}\)

Thus, the correlation coefficient is 0.826.

Step 4:Conduct a hypothesis test for correlation

Define\(\rho \)as the correlation coefficient for the population of two variables, foot lengths and height of males.

For testing the claim, form the hypotheses:

\(\begin{array}{l}{H_o}:\rho = 0\\{H_a}:\rho \ne 0\end{array}\)

The samplesize is 5 (n).

The test statistic is computed as follows:

\(\begin{aligned} t &= \frac{r}{{\sqrt {\frac{{1 - {r^2}}}{{n - 2}}} }}\\ &= \frac{{0.826}}{{\sqrt {\frac{{1 - {{0.826}^2}}}{{5 - 2}}} }}\\ &= 2.538\end{aligned}\)

Thus, the test statistic is 2.538.

The degree of freedom is

\(\begin{aligned} df &= n - 2\\ &= 5 - 2\\ &= 3.\end{aligned}\)

The p-value is computed from the t-distribution table.

\(\begin{aligned} p{\rm{ - value}} &= 2P\left( {T > t} \right)\\ &= 2P\left( {T > 2.538} \right)\\ &= 2\left( {1 - P\left( {T < 2.538} \right)} \right)\\ &= 0.085\end{aligned}\)

Thus, the p-value is 0.085.

Since the p-value is greater than 0.05, the null hypothesis fails to berejected.

Therefore, there is not enough evidence to conclude that foot length and height have a linear correlation between them.

Analyze if the foot length can help predict the height of males

From the above result, the variables foot lengths and height of males are not linearly correlated. On the other hand, the scatterplot shows an upward trend but no specific pattern (linear or non-linear).

Thus, the foot lengths cannot be used to predict heights.