Question

1. Previous studies suggest that certain cell cluster in the brain govern sexual behavior. The volume of a cell cluster, INAH3, in was measured in the interstitial nuclei of the anterior hypothalamus in postmortem tissue of 43 subjects at autopsy from some metropolitan hospitals. A natural log transformation was necessary. Hence, the transformation of

Y =Ln(1000×Volume) was used. All required assumptions are satisfied for the transformed data (the y-values). Subjects were classified into five groups according to three factors: gender (Males, Females), sexual orientation (Heterosexual, Homosexual), and cause of death (AIDS, Non-AIDS). There was no homosexual female and no homosexual male with Non-AIDS as the cause of death in the data.

1. (2 marks) Referring to the output in Tables 1 and 2 below, would you say that there is a significant difference in the average of the y-values between homosexual and heterosexual people? Just refer to the estimated difference and the p-value to answer the question (you don’t have to carry out a formal test).
   Table 1: The summary statistics for y-values between homosexual and heterosexual people are:
   
   Table 2: The output for the comparison of average y-values between homosexual and heterosexual people is:
   
2. (2 marks) To answer the question in part (a), a researcher used the following ANOVA table. How do the test statistics and p-values in tables 2 and 3 compare? Should the p-values be equal? Explain why?
   Table 3: The ANOVA table for the comparison of average y-values between homosexual and heterosexual people is:
   
   Define:
   μ1 : Average of y-values for heterosexual males with AIDS as the cause of the death (HE-M-A)
   μ2 : Average of y-values for heterosexual males with Non-AIDS as the cause of the death (HE-M-NA)
   μ3 : Average of y-values for homosexual males with AIDS as the cause of the death (HO-M-A)
   μ4 : Average of y-values for heterosexual females with AIDS as the cause of the death (HE-F-A)
   μ5 : Average of y-values for heterosexual females with Non-AIDS as the cause of the death (HE-F-NA)
   When needed use Tables 4 to 7 below to answer questions (b) – (g) (you won’t necessarily need all of the tables).
   Table 4: The summary statistics of y-values for the 5 groups are:
   Table 5: The ANOVA table for the comparison of average y-values content for 5 groups:
   
   
3. (4 marks ) Carry out a test to determine if there are any significant differences among the five groups. Clearly indicate the null and alternative hypothesis, the test-statistics and how it was calculated, the null distribution of the test statistic, and the p-value. Interpret the p-value in plain English.
4. (2 marks) What is the best estimate for the common standard deviation of the 5 populations? (Hint: this value can be found (with minor calculation) from a value in the completed ANOVA table. You will need this value for parts (e), (f) and so on.)
5. (15 marks) Construct linear combinations of means to make comparisons among the five group means that will address the following comparisons:

1. Compare the average of y-values for males versus females.
2. Compare the average of y-values for homosexuals versus heterosexual peoples.
3. Compare the average of y-values for people with cause of AIDS death versus people with non-AIDS cause of death.
4. Compare the average of y-values for heterosexual males versus heterosexual females (Ignore homosexual males).
5. Compare the average of y-values for heterosexual males versus homosexual males (Ignore females).

Use the Bonferroni method to construct simultaneous confidence intervals for the above linear combinations with familywise confidence level of 95%. At familywise significance level of 0.05, which linear combinations are significantly different from zero? Which factor (gender, sexual orientation, and/or cause of death) is the most important factor? Explain your idea.

f) (6 marks) Now let’s look at the cause of death effect after accounting for gender. Does there appear to be a significant cause of death effect within each gender? In other words, is there a significant difference in y- values between AIDS cause of death and non-AIDS cause of death with the same gender? Carry out a single overall test to determine if there is a difference in average y-values between people who died because of AIDS and people who died because of non-AIDS who have the same gender (either male or female). In terms of parameters, state clearly the models in the null and alternative hypothesis. Also, identify the sum of squared residuals and degrees of freedom for the models in the null and alternative hypothesis. Calculate the test-statistic, the p-value and identify the distribution of the test-statistic under the null hypothesis. What do you conclude?

Price: $16.33

Solution: The downloadable solution consists of 7 pages, 933 words and 10 charts.
Deliverable: Word Document