# Testing for a Linear Correlation. In Exercises 13–28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find t

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 $\alpha =0.05$. 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.) Lemons and Car Crashes Listed below are annual data for various years. The data are weights (metric tons) of lemons imported from Mexico and U.S. car crash fatality rates per 100,000 population [based on data from “The Trouble with QSAR (or How I Learned to Stop Worrying and Embrace Fallacy),” by Stephen Johnson, Journal of Chemical Information and Modeling, Vol. 48, No. 1]. Is there sufficient evidence to conclude that there is a linear correlation between weights of lemon imports from Mexico and U.S. car fatality rates? Do the results suggest that imported lemons cause car fatalities? $\begin{array}{cccccc}\text{Lemon Imports}& 230& 265& 358& 480& 530\\ \text{Crashe Fatality Rate}& 15.9& 15.7& 15.4& 15.3& 14.9\end{array}$
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Step 1 Note: we are using MINTAB software to perform the calculations. The data shows that the weights of lemon imports from Mexico and U.S. car fatality rates. The level of significance is $$\alpha = 0.05$$. Procedure to obtain scatterplot using the MINITAB software: Choose Graph > Scatterplot. Choose Simple and then click OK. Under Y variables, enter a column of CRASH FERTILITY RATES. Under X variables, enter a column of LEMON IMPORTS. Click OK. Step 2 The hypotheses are given below: Null hypothesis: $$H0:\rho = 0$$ That is, there is no linear correlation between the weights of lemon imports from Mexico and U.S. car fatality rates. Alternative hypothesis: $$H1:\rho\ cancel= 0$$ That is, there is a linear correlation between the weights of lemon imports from Mexico and U.S. car fatality rates. Correlation coefficient r: Software Procedure: Step-by-step procedure to obtain the ‘correlation coefficient’ using the MINITAB software: Select Stat >Basic Statistics > Correlation. In Variables, select LEMON IMPORTS and CRASH FERTILITY RATES from the box on the left. Click OK. Output using the MINITAB software is given below: Correlations: LEMON IMPORTS, CRASH FATALITY RATE Pearson correlation of LEMON IMPORTS and CRASH FATALITY RATE $$= -0.959$$
$$P-value = 0.010$$ Step 3 Thus, the Pearson correlation of weights of lemon imports from Mexico and U.S. car fatality rates is –0.959 and the P-value is 0.010. Critical value: From the TABLE “Critical Values of the Pearson Correlation Coefficient r”, the critical value for 5 degrees of freedom for $$\alpha = 0.05$$ level of significance is $$\pm 0.878$$. The horizontal axis represents weights of lemon imports from Mexico and vertical axis represents U.S. car fatality rates. From the plot, it is observed that there is a linear association between the weights of lemon imports from Mexico and U.S. car fatality rates because the data point show a distinct pattern. The P-value is 0.010 and the level of significance is 0.05. Here, the P-value is less than the level of significance. Hence, the null hypothesis is rejected. That is, there is a linear correlation between the weights of lemon imports from Mexico and U.S. car fatality rates. The critical value is $$\pm 0.878$$. Here, the correlation value –0.959 lies beyond the lower critical value. Thus, there is sufficient evidence to support the claim that there is a linear correlation between the weights of lemon imports from Mexico and U.S. car fatality rates. There is a linear correlation between the weights of lemon imports from Mexico and U.S. car fatality rates but it does not appear the imported lemons cause car fatalities because, it do not suggest any cause-effect relationship.