Question

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then the conclusion about the null hypothesis, as well as the final conclusion that address the original claim. Assume that a simple random sample is selected from a normally distributed population.

Pulse Rates of Men: A simple random sample of 153 men results in a standard deviation of 11.3 beats per minute (based on Data Set 1 “Body Data” in Appendix B). The normal range of pulse rates of adults is typically given as 60 to 100 beats per minute. If the range rule of thumb is applied to that normal range, the result is a standard deviation of 10 beats per minute. Use the sample results with a 0.05 significance level to test the claim that pulse rates of men have a standard deviation equal to 10 beats per minute; see the accompanying StatCrunch display for this test. What do the results indicate about the effectiveness of using the range rule of thumb with the “normal range” from 60 to 100 beats per minute for estimating s in this case?

Short answer

The hypotheses are formulated as follows.

$$\begin{gathered} H_0:\sigma =10\text{bpm} \\ {\text{H}}_{\text{1}}:\sigma \ne \text{10 bpm} \end{gathered}$$

The test statistic is ${\chi }^2=195.17$, and the P-value = 0.0208. Reject ${\text{H}}_{\text{0}}$.

There is not enough evidence to support the claim that the standard deviation of the pulse rates is 10 bpm.

The range rule of thumb does not give a very good estimate of the standard deviation.

Step-by-step solution

Given information 

The standard deviation of the pulse rates for a sample of 153 men is 11.3 beats per minute. The range rule of thumb estimate of the standard deviation of the population of the mean is 10 beats per minute.

State the hypotheses

To test the significance of theclaim that pulse rates of men have a standard deviation equal to 10 beats per minute, the null and alternative hypotheses are formulated as follows.

$$\begin{gathered} H_0:\sigma =10 \\ {H}_1:\sigma \ne 10 \end{gathered}$$

Here, $\sigma$is the actual standard deviation of the pulse rates for the population of men.

State the test statistic 

From the table, the test statistic is 195.1723, and the degree of freedom is 152, obtained from the third and fourth columns, respectively.

The P-value for the test is obtained from the fifth column as 0.0208.

State the decision rule

The decision rule states the following:

If the P-value is greater than 0.05, the null hypothesis is failed to be rejected; otherwise, it is rejected.

In this case, the P-value is lesser than 0.05. Therefore, reject $H_0$.

Conclusion of the test

As the null hypothesis is rejected, it can be concluded that there is insufficient evidence to support the claim that the pulse rates of adults have a standard deviation equal to 10.

The hypothesized value of 10 beats per minute is estimated using the range rule of thumb with the normal range of 60 to 100 beats per minute. As the true value of the population standard deviation is not equal to 10 bpm, it can be said that the estimate is not a very good estimate for in this case.