91影视

Association between two variables, pizza cost and subway fare,isbeing studied.

A plot that shows observations from two variablesby scaling them on two axes is referred to as a scatterplot.

Steps to sketch a scatterplot:

1. Mark horizontal axis for price cost and vertical axis for subway fare.
2. Mark points for each paired value with respect to both axes.

The resultant graph is the required scatterplot.

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 pizza cost be defined by variablex and subway fare be defined by variabley.

The valuesare listed in the table below:

Substitute the values in the formula:

\(\begin{aligned} r &= \frac{{9\left( {27.8325} \right) - \left( {13.8} \right)\left( {13.85} \right)}}{{\sqrt {9\left( {27.685} \right) - {{\left( {13.8} \right)}^2}} \sqrt {9\left( {28.0925} \right) - {{\left( {13.85} \right)}^2}} }}\\ &= 0.992\end{aligned}\)

Thus, the correlation coefficient is 0.992.

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 is 9 (n).

The test statistic is computed as follows:

\(\begin{aligned} t &= \frac{r}{{\sqrt {\frac{{1 - {r^2}}}{{n - 2}}} }}\\ &= \frac{{0.992}}{{\sqrt {\frac{{1 - {{0.992}^2}}}{{9 - 2}}} }}\\ &= 20.791\end{aligned}\)

Thus, the test statistic is 20.791.

The degree of freedom is

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

Thep-value is computed from the t-distribution table.

\(\begin{aligned} p{\rm{ - value}} &= 2P\left( {T > t} \right)\\ &= 2P\left( {T > 20.791} \right)\\ &= 2\left( {1 - P\left( {t < 20.791} \right)} \right)\\ &= 0.000\end{aligned}\)

Thus, the p-value is 0.000.

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

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