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.)

Old Faithful Listed below are duration times (seconds) and time intervals (min) to the next eruption for randomly selected eruptions of the Old Faithful geyser in Yellowstone National Park. Is there sufficient evidence to conclude that there is a linear correlation between duration times and interval after times?

Duration	242	255	227	251	262	207	140
Interval After	91	81	91	92	102	94	91

The data is recorded for two variables: duration in seconds and time intervals in minutes for the next eruption of a geyser.

A scatterplot is a graph oftwo variables that havepaired values. Each variable is scaled on one axis.

Steps to sketch a scatterplot:

1. Mark two axes, xand y,for duration and interval after, respectively.
2. Mark the paired data values on the graph corresponding to the axes.

The resultant graph is shown below.

The formula for the 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 the duration be variable x and theinterval after be variable y.

The valuesare listedin the table below:

Substitute the values in the formula:

\(\begin{aligned} r &= \frac{{7\left( {145348} \right) - \left( {1584} \right)\left( {642} \right)}}{{\sqrt {7\left( {369212} \right) - {{\left( {1584} \right)}^2}} \sqrt {7\left( {59108} \right) - {{\left( {642} \right)}^2}} }}\\ &= 0.046\end{aligned}\)

Thus, the correlation coefficient is 0.046.

Define\(\rho \)as the true measure ofthe correlation coefficient for the two variables.

For testing the claim, form the hypotheses:

\(\begin{array}{l}{{\rm{H}}_{\rm{o}}}:\rho = 0\\{{\rm{{\rm H}}}_{\rm{a}}}:\rho \ne 0\end{array}\)

The samplesize is7 (n).

The test statistic is computed as follows:

\(\begin{aligned} t &= \frac{r}{{\sqrt {\frac{{1 - {r^2}}}{{n - 2}}} }}\\ &= \frac{{0.046}}{{\sqrt {\frac{{1 - {{0.046}^2}}}{{7 - 2}}} }}\\ &= 0.103\end{aligned}\)

Thus, the test statistic is 0.103.

The degree of freedom is

\(\begin{aligned} df &= n - 2\\ &= 7 - 2\\ &= 5.\end{aligned}\)

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

\(\begin{aligned} p{\rm{ - value}} &= 2P\left( {T > t} \right)\\ &= 2P\left( {T > 0.103} \right)\\ &= 2\left( {1 - P\left( {T < 0.103} \right)} \right)\\ &= 0.921\end{aligned}\)

Thus, the p-value is 0.921.

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

Therefore, there is not enough evidence to conclude that variables x(duration) and y (interval after) have a linear correlation between them.