Question: Blood Pressure. A person's systolic blood pressure can be a signal of serious concerns in their cardiovascular system. A study collected data on 500 randomly selected adults. Of interest is the systolic blood pressure and how it relates to the weight of a person. Weight is a categorical variable taking on the values \(0=\) Normal, \(1=\) Overweight and \(2=\) Obese. The variable name is Overwt. The primary research question is if the mean systolic blood pressure is different for these three groups of people at the \(\alpha=0.05\) level of significance. To answer each of the following questions, refer to the output in Figure 1 .

1. Why should we not use two-sample t-tests to see what differences there are between the means of these three groups.
2. One-way ANOVA requires homogeneous (constant) variances across all groups. Apply the rule of thumb to assess whether the constant variance assumption is at least approximately satisfied. Show all your work.
3. Conduct the appropriate test to answer the primary research question. Be sure to include the null- and alternative hypothesis, test statistic, p-value, decision and conclusion within the context of the research question.
4. Based on your findings in (c), can we conclude that at least one of the two groups (overweight or obese) causes, on average, higher systolic blood pressure? Explain why or why not.
5. Regardless of your conclusion in part (c), we decide to follow up with considering all pairwise comparisons. If we apply the Bonferroni adjustment, to what value will the current a = 0.05 level of significance change?
6. Regardless of your answer in part (3e), assume the adjusted level of significance is now \(\alpha=0.01\). Which of the group means are significantly different at the \(\alpha=0.01\) level? With your choice also state the estimated difference in the average systolic blood pressure.
   - Obese versus Normal:
   \(\begin{array}{lll}\text { SIGNIFICANT } & \text { NOT SIGNIFICANT, } & \text { estimated difference: }\end{array}\)
   - Obese versus Overweight:
   \(\begin{array}{lll}\text { SIGNIFICANT } & \text { NOT SIGNIFICANT, } & \text { estimated difference: }\end{array}\)
   - Overweight versus Normal:
   SIGNIFICANT \(\quad\) NOT SIGNIFICANT, estimated difference:
7. Test the following hypothesis by setting up the appropriate contrast. Use \(\alpha=0.01\). On average, people that were considered obese have higher systolic blood pressure than people who are considered Normal or overweight.

1. State the appropriate null and alternative hypothesis.
2. Write out how the contrast \(\psi\) is defined.
3. What are the values of the contrast coefficients \(a_{\text {normal }}, a_{\text {overweight }}\) and \(a_{\text {obese }} ?\)
   \(a_{\text {Normal }}=a_{\text {Overweight }}=a_{\text {Obese }}=\)
4. From JMP, report the corresponding estimate of the contrast and its standard error.
   c:
   \(S E_{c}:\)
5. Report the test statistic and the p-value and make a decision about the null hypothesis.
   test statistic:
   \(\mathrm{p}\) -value:
   \(\begin{array}{ll}\text { decision: } & \text { REJECT } \mathrm{H}_{0}\end{array}\)
   FAIL TO REJECT H \(_{0}\)
6. Write a conclusion in the context of the problem.

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