A medical researcher wants to determine whether male pulse rates vary more or less than female pulse rates. Using the sample pulse rates in Data Set 1 in Appendix B, the male pulse rates can be
summarized with the statistics
40=n,69.4=x,11.3=s.
For females, the statistics are,40=n,76.3=x, 12.5=s Assume that each sample is a simple random sample obtained from a population
with a normal distribution.
a) Construct a 90% confidence interval estimate of the population standard deviation σ of pulse rates for males.
b) Construct a 90% confidence interval estimate of the population standard deviation of pulse rates for females.
c) Compare the preceding results.
Does it appear that the population standard deviations for males and females are different? Why or why not?
To construct confidence interval estimates for the population standard deviations of pulse rates for males and females, we can use the formula:
Confidence Interval = (sample measure) ± (critical value) * (standard error)
a) To construct a 90% confidence interval estimate of the population standard deviation for males, we use the sample statistics given: n = 40, x = 69.4, and s = 11.3.
1. Find the critical value corresponding to a 90% confidence level in a chi-square distribution table. For a 90% confidence level and 39 degrees of freedom (n-1), the critical values are approximately 24.4336 (lower) and 57.4896 (upper).
2. Calculate the standard error using the formula: standard error = s / sqrt(n)
Substituting values: standard error = 11.3 / sqrt(40) ≈ 1.7874.
3. Construct the interval:
Confidence Interval = x ± (critical value) * (standard error)
Confidence Interval = 69.4 ± 24.4336 * 1.7874
Confidence Interval ≈ (66.46, 72.34)
b) To construct a 90% confidence interval estimate of the population standard deviation for females, we use the sample statistics given: n = 40, x = 76.3, and s = 12.5.
1. Find the critical value corresponding to a 90% confidence level in a chi-square distribution table (same as step 1 in part a).
2. Calculate the standard error using the formula: standard error = s / sqrt(n)
Substituting values: standard error = 12.5 / sqrt(40) ≈ 1.9799.
3. Construct the interval:
Confidence Interval = x ± (critical value) * (standard error)
Confidence Interval = 76.3 ± 24.4336 * 1.9799
Confidence Interval ≈ (71.35, 81.25)
c) To compare the results, we can analyze the confidence intervals. Since the confidence intervals for males and females do not overlap, it suggests that their population standard deviations are likely different.
In conclusion, based on the 90% confidence intervals, it appears that the population standard deviations for males and females are different. However, further statistical analyses and hypothesis testing may be required to confirm this observation.