91Ó°ÊÓ

The data is recorded forchirps of crickets and temperatures in degrees Fahrenheit.

Scatterplot projects a paired set of observationsontwo axes scaled for the two variables.

Steps to sketch a scatterplot:

1. Describe two axes, x and y, for chirps in 1 minute and temperature, respectively.
2. Mark the points on the graph.

The graph is shown below.

The correlation coefficient formula is

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

Describe variables x and y as chirps in 1 minute and temperature, respectively.

The valuesare listed in the table below:

Substitute the values in the formula:

\(\begin{aligned} r &= \frac{{8\left( {663245.4} \right) - \left( {8130} \right)\left( {646} \right)}}{{\sqrt {8\left( {8391204} \right) - {{\left( {8130} \right)}^2}} \sqrt {8\left( {52626.6} \right) - {{\left( {646} \right)}^2}} }}\\ &= 0.874\end{aligned}\)

Thus, the correlation coefficient is 0.874.

Definethe actual measure of the correlation coefficient between chirps and temperature as\(\rho \).

For testing the claim, form the hypotheses:

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

The samplesize is 8(n).

The test statistic is computed as follows:

\(\begin{aligned} t &= \frac{r}{{\sqrt {\frac{{1 - {r^2}}}{{n - 2}}} }}\\ &= \frac{{0.874}}{{\sqrt {\frac{{1 - {{\left( {0.874} \right)}^2}}}{{8 - 2}}} }}\\ &= 4.406\end{aligned}\)

Thus, the test statistic is 4.406.

The degree of freedom is

\(\begin{aligned} df &= n - 2\\ &= 8 - 2\\ &= 6.\end{aligned}\)

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

\(\begin{aligned} p{\rm{ - value}} &= 2P\left( {t > 4.406} \right)\\ &= 0.0045\\ &\approx 0.005\end{aligned}\)

Thus, the p-value is 0.005.

Since thep-value is lesser than 0.05, the null hypothesis is rejected.

Therefore, there is enough evidence to conclude a linear correlation between chirps in 1 minute and temperature.