Chapter 16, Problem

To determine

State the hypotheses, test statistic and the two-tailed p-value.

Make a decision for the research question “whether there is significant correlation between gasoline price and carbon dioxide emission?” and identify whether the research question is close to the obtained decision.

Identify the issues of sample size.

Identify whether non-normality is concerned or not.

Answer

The hypotheses for the test are given below:

Null hypothesis:

$H_0:{\rho }_s\le 0$

The rank correlation between gasoline price and carbon dioxide emission is zero.

Alternate Hypothesis:

$H_1:{\rho }_s>0$

The rank correlation between gasoline price and carbon dioxide is greater than zero.

The test statistic and the p-value are –0.588 and 0.355 respectively.

There is a support evidence to conclude that there exists a significant correlation between gasoline price and carbon dioxide emission at 5% level of significance.

The decision is close since the rejection of null hypothesis is close to the research question “whether there is significant correlation between gasoline price and carbon dioxide emission?”

The sample size is not an issue.

The gasoline price follows normality but the carbon dioxide emission does not follow normality.

Explanation of Solution

Calculation:

The given information is that, the data shows gasoline price and carbon emissions for selected nations. The level of significance is 0.05.

The hypotheses for the test are given below:

Null hypothesis:

$H_0:{\rho }_s\le 0$

The rank correlation between gasoline price and carbon dioxide emission is zero.

Alternate Hypothesis:

$H_1:{\rho }_s>0$

The rank correlation between gasoline price and carbon dioxide emission is greater than zero.

Software procedure:

Step-by-step procedure to find the Spearman rank correlation using MINITAB is given below:

• Choose MegaStat > Nonparametric Tests > Spearman Coefficient of Rank Correlation.
• In Input range, select the cells A1:B32.
• Unselect Output ranked data and Correct for ties.
• Click OK.

Output obtained from MegaStat is given below:

Decision Rule:

Reject the null hypothesis $\left(H_0\right)$ when the p-value is lesser than or equal to the level of significance $\alpha$, otherwise do not reject the null hypothesis $\left(H_0\right)$.

Conclusion:

The absolute value of the test statistic is 0.588 and the critical value for the desired level of significance $\alpha$ is $0.355$.

The test statistic is greater than the critical value.

That is, $0.588\left(=\text{test statistic}\right)>0.355\left(=\text{critical value}\right)$.

Thus, the null hypothesis is rejected.

Hence, there is a support of evidence to conclude that there exists a significant correlation between gasoline price and carbon dioxide at 5% level of significance.

Histogram for gasoline price:

Step-by-step procedure to construct a histogram using MINITAB is given below:

• Choose Basic Statistics > Graphical Summary.
• Choose Simple, and then click OK.
• In variables, enter the column of Gasoline price.
• In Confidence level, enter 95.0.
• Click OK.

Output obtained from MINITAB is given below:

Interpretation:

The histogram appears to be left skewed since the tail is elongated towards the left than to the right side. Hence, the test of normality is recommended.

Testing the normality using Anderson Darling test:

Null hypothesis:

$H_0:$ The given data follows normal distribution.

Alternate Hypothesis:

$H_1:$ The given data do not follow normal distribution.

Decision Rule:

Reject the null hypothesis $\left(H_0\right)$ when the p-value is lesser than or equal to the level of significance $\alpha$, otherwise do not reject the null hypothesis $\left(H_0\right)$.

Conclusion:

The p-value for the A-D test is 0.169 and the level of significance is 0.05.

The p-value for the A-D test is greater than the level of significance.

That is, $0.169\left(=p\text{-value}\right)>0.05\left(=\alpha \right)$

Thus, the null hypothesis is rejected.

Hence, there is a support of evidence to assume that gasoline follows normal distribution at 5% level of significance.

Histogram for carbon dioxide emission:

Step-by-step procedure to construct a histogram using MINITAB is given below:

• Choose Basic Statistics > Graphical Summary.
• Choose Simple, and then click OK.
• In variables, enter the column of carbon dioxide emission.
• In Confidence level, enter 95.0.
• Click OK.

Output obtained from MINITAB is given below:

Interpretation:

The histogram appears to be right skewed since the tail is elongated towards the right than to the left side. Hence, the test of normality is recommended.

Testing the normality using Anderson Darling test:

Null hypothesis:

$H_0:$ The given data follows normal distribution.

Alternate Hypothesis:

$H_1:$ The given data do not follow normal distribution.

Decision Rule:

Reject the null hypothesis $\left(H_0\right)$ when the p-value is lesser than or equal to the level of significance $\alpha$, otherwise do not reject the null hypothesis $\left(H_0\right)$.

Conclusion:

The p-value for the A-D test is lesser than 0.005 and the level of significance is 0.05.

The p-value for the A-D test is lesser than the level of significance.

That is, $<0.005\left(=p\text{-value}\right)<0.05\left(=\alpha \right)$

Thus, the null hypothesis is rejected.

Hence, there is no support of evidence to assume that the carbon dioxide emission follows normal distribution at 5% level of significance.