Question 26, 8.4.5
Part 4 of 4
HW Score: $78.13 \%, 21.88$ of
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In a study of pulse rates of men, a simple random sample of 144 men results in a standard deviation of 11.0 beats per minute. 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 0.01 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 $\sigma$ in this case? Assume that the simple random sample is selected from a normally distributed population.
(i) Click the icon to view the StatCrunch display.
(Round to three decimal places as needed)
State the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim
the null hypothesis. There sufficient evidence to the claim that pulse rates of men have a standard deviation equal to 10 beats per minute. The results indicate that there significant evidence that using the range rule of thumb with the "normal range" from 60 to 100 beats per minute for estimating $\sigma$ effective in this case.
Answer
Final Answer: There is not sufficient evidence to reject the claim that pulse rates of men have a standard deviation equal to 10 beats per minute. The results indicate that using the range rule of thumb with the 'normal range' from 60 to 100 beats per minute for estimating \(\sigma\) is effective in this case. Therefore, the final conclusion is \(\boxed{\text{Do not reject the null hypothesis}}\).
Step 1 ：The problem is asking to test the claim that pulse rates of men have a standard deviation equal to 10 beats per minute. The sample size is 144 and the sample standard deviation is 11.0 beats per minute. The significance level is 0.01. We can use the chi-square test for a single variance to test this claim. The null hypothesis is that the population standard deviation is equal to 10 beats per minute. The alternative hypothesis is that the population standard deviation is not equal to 10 beats per minute.
Step 2 ：Given values are: sample size (n) = 144, sample standard deviation (s) = 11.0, population standard deviation (sigma) = 10, significance level (alpha) = 0.01.
Step 3 ：Calculate the chi-square statistic using the formula: \(\chi^2 = (n-1)\frac{s^2}{\sigma^2}\). Substituting the given values, we get \(\chi^2 = 173.03\).
Step 4 ：Calculate the critical chi-square values for the left and right tails of the distribution using the chi-square distribution table. The critical values are \(\chi^2_{left} = 103.19555083328622\) and \(\chi^2_{right} = 190.3063753701604\).
Step 5 ：The test statistic is greater than the critical value on the left and less than the critical value on the right. Therefore, we do not reject the null hypothesis. This means that there is not enough evidence to support the claim that the population standard deviation is not equal to 10 beats per minute.
Step 6 ：Final Answer: There is not sufficient evidence to reject the claim that pulse rates of men have a standard deviation equal to 10 beats per minute. The results indicate that using the range rule of thumb with the 'normal range' from 60 to 100 beats per minute for estimating \(\sigma\) is effective in this case. Therefore, the final conclusion is \(\boxed{\text{Do not reject the null hypothesis}}\).
