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.)

CPI and the Subway Use CPI>subway data from the preceding exercise to determine whether there is a significant linear correlation between the CPI (Consumer Price Index) and the subway fare.

Short answer

The scatter plot is shown below:

The value of the correlation coefficient is 0.973.

The p-value is 0.000.

There is enough evidence to support the claim that there is a linear correlation between the two variables (CPI and subway fare).

Step-by-step solution

Given information

Refer to Exercise 15 for the data.

Year	1960	1973	1986	1995	2002	2003	2009	2013	2015
Pizza Cost	0.15	0.35	1	1.25	1.75	2	2.25	2.3	2.75
Subway Fare	0.15	0.35	1	1.35	1.5	2	2.25	2.5	2.75
CPI	30.2	48.3	112.3	162.2	191.9	197.8	214.5	233	237.2

Sketch a scatterplot

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.

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

The valuesare listedin the table below:

x	y	\({x^2}\)	\({y^2}\)	\(xy\)
30.2	0.15	912.04	0.0225	4.53
48.3	0.35	2332.9	0.1225	16.905
112.3	1	12611	1	112.3
162.2	1.35	26309	1.8225	218.97
191.9	1.5	36826	2.25	287.85
197.8	2	39125	4	395.6
214.5	2.25	46010	5.0625	482.63
233	2.5	54289	6.25	582.5
237.2	2.75	56264	7.5625	652.3
\(\sum x = 1427.4\)	\(\sum y = 13.85\)	\(\sum {{x^2}} = 274678.6\)	\(\sum {{y^2} = } 28.0925\)	\(\sum {xy\; = \;} 2753.58\)

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.

Step 4:Conduct a hypothesis test for correlation

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.