Question

Clinical researchers wish to see if hypertension increases the aortic stiffness index (calculated from the aortic diameter evaluated by M-mode echocardiography and blood pressure measured by a sphygmomanometer) in humans.

They take a sample of 31 subjects with hypertension and compute their mean aortic stiffness index to be 18.46 with a standard deviation of 4.9. While a sample of 21 healthy subjects gives a mean aortic stiffness index of 12.63 and a standard deviation of 2.83.

Using a significance level of α=0.05, test the claim that people who have hypertension have a higher aortic stiffness index on average than people without hypertension.

A) the critical value is...

B) the test value is...

Answer 1

To test the claim that people with hypertension have a higher aortic stiffness index than people without hypertension, we can perform a two-sample t-test. The critical value is approximately 2.01. The test value is approximately 6.46.

Step-by-step explanation:

To test the claim that people with hypertension have a higher aortic stiffness index on average than people without hypertension, we can perform a two-sample t-test. The null hypothesis is that there is no difference between the means of the two groups. The alternative hypothesis is that the mean aortic stiffness index of the hypertension group is higher than that of the healthy group.

Using a significance level of α = 0.05, we can calculate the critical value and the test value. The critical value is the value that separates the rejection region from the non-rejection region. In this case, the critical value corresponds to the 5% level of significance and can be found using a t-distribution table or calculator.

The test value, also known as the t-statistic, measures how far the sample mean is from the hypothesized population mean. It is calculated using the formula:

t = (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2))

Where:

x1 = mean aortic stiffness index of the hypertension group

x2 = mean aortic stiffness index of the healthy group

s1 = standard deviation of the hypertension group

s2 = standard deviation of the healthy group

n1 = sample size of the hypertension group

n2 = sample size of the healthy group

Using the given information:

x1 = 18.46, x2 = 12.63

s1 = 4.9, s2 = 2.83

n1 = 31, n2 = 21

Using a t-distribution table or calculator, the critical value for a two-tailed test with α = 0.05 and degrees of freedom df = n1 + n2 - 2 = 31 + 21 - 2 = 50 is approximately t = 2.01.

Calculating the test value using the given formula: t = (18.46 - 12.63) / sqrt((4.9^2 / 31) + (2.83^2 / 21)) ≈ 6.46

Since the test value (6.46) is greater than the critical value (2.01), we reject the null hypothesis. We have sufficient evidence to support the claim that people with hypertension have a higher aortic stiffness index on average than people without hypertension.