Question 13
2 pts
A researcher wants to assess if there is a difference in the average age of onset of a type of kidney disease for men and women. Let the population 1 mean be the average age of onset for women and the population 2 mean be the average age of onset for men. A random sample of 27 women with the disease showed an average age of onset to be 81 years, with a sample standard deviation of 13.5 years. A random sample of 22 men with the disease showed an average age of onset to be 79 years with a sample standard deviation of 6.5 years. Assume that ages at onset of this disease are normally distributed for each gender also assume unequal variance. Conduct a hypothesis test at the .05 level of significance to determine if there is a difference in the onset ages between men and women. What is/are the critical value(s)?

Step 1 ：Define the null hypothesis (H0) as there being no difference in the average age of onset for men and women, and the alternative hypothesis (H1) as there being a difference.

Step 2 ：Use the formula for the t statistic in Welch's t-test: \(t = \frac{{X1 - X2}}{{\sqrt{{(s1^2/n1) + (s2^2/n2)}}}}\)

Step 3 ：Substitute the given values into the formula: \(t = \frac{{81 - 79}}{{\sqrt{{(13.5^2/27) + (6.5^2/22)}}}} = \frac{2}{{\sqrt{{(182.25/27) + (42.25/22)}}}} = \frac{2}{{\sqrt{{6.75 + 1.92}}}} = \frac{2}{{\sqrt{{8.67}}}} = \frac{2}{2.94} = 0.68\)

Step 4 ：Approximate the degrees of freedom for this test using the Welch-Satterthwaite equation: \(df = \frac{{(s1^2/n1 + s2^2/n2)^2}}{{((s1^2/n1)^2/(n1-1)) + ((s2^2/n2)^2/(n2-1))}}\)

Step 5 ：Substitute the given values into the formula: \(df = \frac{{(182.25/27 + 42.25/22)^2}}{{((182.25/27)^2/(27-1)) + ((42.25/22)^2/(22-1))}} = \frac{{8.67^2}}{{(6.75^2/26) + (1.92^2/21)}} = \frac{{75.21}}{{1.75 + 0.18}} = \frac{{75.21}}{{1.93}} = 39\)

Step 6 ：Find the critical value(s) for a two-tailed t-test at the .05 level of significance with 39 degrees of freedom. Using a t-distribution table or calculator, the critical values are approximately ±2.021.

Step 7 ：So, the critical values for this hypothesis test are approximately ±2.021. \(\boxed{\pm 2.021}\)