91影视

Refer to Exercise 15 for the data.

A scatterplot hasdots torepresent paired observations of a dataset projected corresponding to theaxes scaled for two variables.

Steps to sketch a scatterplot:

1. Mark horizontal axis for CPI and vertical axis for subway fare.
2. Mark the points ofobservations corresponding to each axis.
3. The resultant graph is the scatterplot.

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 CPI be defined by variable x and subway fare be defined by variable y.

The valuesare listedin the table below:

Substitute the values in the formula:

\(\begin{aligned}{c}r &= \frac{{9\left( {2753.58} \right) - \left( {1427.4} \right)\left( {13.85} \right)}}{{\sqrt {9\left( {274678.6} \right) - {{\left( {1427.4} \right)}^2}} \sqrt {9\left( {28.0925} \right) - {{\left( {13.85} \right)}^2}} }}\\ &= 0.973\end{aligned}\)

Thus, the correlation coefficient is 0.973.

Define\(\rho \)as the actual value of thecorrelation coefficient for pizza cost and subway fare.

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 is9 (n).

The test statistic is computed as follows:

\(\begin{aligned} t &= \frac{r}{{\sqrt {\frac{{1 - {r^2}}}{{n - 2}}} }}\\ &= \frac{{0.973}}{{\sqrt {\frac{{1 - {{0.973}^2}}}{{9 - 2}}} }}\\ &= 11.154\end{aligned}\)

Thus, the test statistic is 11.154

The degree of freedom is

\(\begin{aligned} df &= n - 2\\ &= 9 - 2\\ &= 7.\end{aligned}\)

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

\(\begin{aligned}{c}p{\rm{ - value}} &= 2P\left( {T > t} \right)\\ &= 2P\left( {T > 11.154} \right)\\ &= 2\left( {1 - P\left( {T < 11.154} \right)} \right)\\ &= 0.000\end{aligned}\)

Thus, the p-value is 0.000.

Since the p-value is less than 0.05, the null hypothesis is rejected.

Therefore, there is enough evidence to conclude that the variables CPI and subway fare have a linear correlation between them.