1. The assessment of diet is an important exposure for many disease outcomes. However, there is often a lot of imprecision in dietary recall. In one study, 20-40 year old mothers of preschool children aged 2-4 were asked about the average servings of margarine per week. Two different recording methods were used to capture servings per week on 12 mothers. The first recording method was the food frequency questionnaire (FFQ) and the second was the real time dietary record (RDR). The table below provides the data for these two methods of collecting data on margarine intake (servings per week). Use these data to answer the questions below the table

ID	FFQ Margarine Intake (servings/week)	RDR Margarine Intake (servings/week)
1	7	0
2	7	0.5
3	0	0
4	0	0
5	0	0
6	7	2.5
7	7	3
8	7	0.5
9	21	3.7
10	0	2.5
11	21	4.1
12	7	8.5

a. Assume that FFQ and RDR intake data are normally distributed. Test the hypothesis that the true Pearson correlation () is significantly different from zero. Show the 5-step hypothesis testing procedure and assume =0.05,

b. What is the p-value for the test statistic you obtained in question a?

c. Using regression methods, find the regression equation that uses FFQ data on margarine intake to predict RDR margarine intake. Do the results of your regression analysis indicate that margarine intake measured by the FFQ is a good predictor of RDR margarine intake? Provide at least one statistic from the output to support your conclusion.

d. Are the results from your regression analyses surprising, given the results you obtained for the Pearson's correlation coefficient in questions a and b? Explain.