Question

A study at the University of Colorado at Boulder shows that running increases the percent resting metabolic rate (RMR) in older women. The average RMR of 30 elderly women runners was 34% higher than the average RMR of 30 sedentary (inactive) elderly women and the standard deviation were reported to be 10.5 % and 10.2 %, respectively. Was there a significant increase in RMR of the women runners over the sedentary women? Use a as 0.05 level of significant, assume the population to be approximately normally distributed with equal variances. Write the steps like the following: Step 1: H₀ and H₁ Step 2:a Step 3: Test statistic Step 4: Show the critical region on the curve Step 5: Conclusion

Answer 1

To determine if there was a significant increase in RMR of the women runners over the sedentary women, we need to conduct a hypothesis test.

Step-by-step explanation:

To determine if there was a significant increase in RMR of the women runners over the sedentary women, we need to conduct a hypothesis test.

Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁).
H₀: There is no significant difference in RMR between the women runners and sedentary women.
H₁: There is a significant increase in RMR of the women runners over the sedentary women.

Step 2: Set the significance level (a) to 0.05.

Step 3: Calculate the test statistic.
We can use a two-sample t-test since we have two independent samples and want to compare their means.
The test statistic formula is:
t = (mean1 - mean2) / sqrt((s1^2 / n1) + (s2^2 / n2))
where mean1 is the mean RMR of the women runners, mean2 is the mean RMR of the sedentary women, s1 is the standard deviation of the women runners, s2 is the standard deviation of the sedentary women, n1 is the sample size of the women runners, and n2 is the sample size of the sedentary women.

Step 4: Determine the critical region in the t-distribution.
In this case, we want to check if the test statistic falls in the critical region corresponding to the significance level of 0.05 in the t-distribution with (n1 + n2 - 2) degrees of freedom.

Step 5: Make a conclusion.
If the test statistic falls in the critical region, we reject the null hypothesis and conclude that there is a significant increase in RMR of the women runners over the sedentary women. Otherwise, if the test statistic does not fall in the critical region, we fail to reject the null hypothesis and cannot conclude a significant increase in RMR.

Answer 2

To investigate if running significantly increases RMR in older women, a two-sample t-test for independent samples is used with an alpha level set at 0.05. The null hypothesis stating no increase in RMR is tested against an alternative hypothesis of a positive increase. The conclusion is based on whether the calculated t-value falls within the specified critical region.

Step-by-step explanation:

Steps for Hypothesis Testing

The question at hand is whether running increases the percent resting metabolic rate (RMR) in older women significantly. To proceed with a hypothesis test to examine this claim, the following steps are taken:

1. Step 1: H₀ and H₁
   H₀: The null hypothesis (H₀) is that there is no difference in RMR between elderly women runners and sedentary elderly women, meaning the increase in RMR due to running is 0 or negative.
   H₁: The alternative hypothesis (H₁) is that there is an increase in RMR for elderly women runners compared to sedentary elderly women, meaning the increase is positive.
2. Step 2: α
   The level of significance (α) is set at 0.05, which is the probability of rejecting the null hypothesis when it is true.
3. Step 3: Test statistic
   We would use a two-sample t-test for independent samples since we have two separate groups and are assuming equal variances.
4. Step 4: Critical region on the curve
   The critical region on the t-distribution curve would be determined by the t-score that corresponds to the 0.05 significance level for a two-tailed test, as we are looking for an increase.
5. Step 5: Conclusion
   We would compare the calculated t-value to the critical t-value. If the calculated t-value falls within the critical region, we reject the null hypothesis, indicating a significant increase in RMR. Otherwise, we fail to reject the null hypothesis.