Question

Technology. In Exercises 9–12, test the given claim by using the display provided from technology. Use a 0.05 significance level. Identify the null and alternative hypotheses, test statistic, P-value (or range of P-values), or critical value(s), and state the final conclusion that addresses the original claim.

Tornadoes Data Set 22 “Tornadoes” in Appendix B includes data from 500 random tornadoes. The accompanying StatCrunch display results from using the tornado lengths (miles) to test the claim that the mean tornado length is greater than 2.5 miles.

Short answer

The hypotheses are as follows.

\[\begin{array}{l}{H_0}:\mu = 2.5\\{H_1}:\mu > 2.5\end{array}\]

The test statistic is 0.7586.

The p-value is 0.2242.

The null hypothesis is failed to be rejected.

Thus, it is concluded that there is not sufficient evidence to support the claim that the tornado length is greater than 2.5 miles.

Step-by-step solution

Given information

A sample is taken from the tornado length with a sample size of 500 with the claim that the population mean of the tornado length is greater than \[2.5\,\,{\rm{miles}}\].

State the hypotheses

The null hypothesis\[{H_0}\]represents the population mean of the tornado length, which is equal to\[2.5\,\,{\rm{miles}}\]. As equality is not present in the claim, the alternate hypothesis\[{H_1}\]represents the population mean of the tornado length, which is greater than\[2.5\,\,{\rm{miles}}\].

Let\[\mu \]be the population mean of the tornado length in miles.

State the null and alternate hypotheses.

\[\begin{array}{l}{H_0}:\mu = 2.5\\{H_1}:\mu > 2.5\end{array}\].

State the test statistic and the p-value from the given output

State the test statistic (T-stat) and the p-value obtained from the fifth column and the sixth column of the given output, respectively.

\[\begin{array}{c}t = 0.75865215\\ \approx 0.7586\\p = 0.2242\\ \approx 0.2242\end{array}\]

State the decision rule for the p-value

Reject the null hypothesis when the p-value is less than the significance level. Otherwise, fail to reject the null hypothesis.

Suppose the significance level is equal to\[0.05\].

The p-value is greater than . Thus, the null hypothesis is failed to be rejected at a 0.05 level of significance.

Conclusion

As the null hypothesis is failed to be rejected, it can be concluded that there is not sufficient evidence to support the claim that the population mean of the tornado length is greater than \[2.5\,\,{\rm{miles}}\].