A nutritionist is interested in studying the relationship between age and muscle

Question

A nutritionist is interested in studying the relationship between age and muscle mass in women of age 40. He obtains data on 60 women of age 40-79 randomly selected from his hospital’s database. Since muscle mass and age are quantitative variables, he decides to use simple linear regression to model the relationship. Because he thinks muscle mass declines with age, he uses age as the explanatory variable and muscle mass as the response variable.

(1 mark) Is this an observational study or experiment?

(1 mark) Based on the scenario, what is the target population for this study? What is the study population?

(1 mark) How might the differences between the study population and target population affect the results of the conclusions about the relationship between age and muscle mass?

Use the following output from R to answer the questions in (d) – (f)

(1 mark) What are the slope and intercept of the least squares regression line for predicting muscle mass from age?

(1 mark) What is the value of the residual sum of squares (SSE)?

(1 mark) What is the size of the typical error when predicting muscle mass from age?

Question 2: Muscle Mass Study – continued (8 Marks)

(3 marks) Use the scenario from Question 1 and the plots above to determine whether the conditions for regression have been adequately met. Explain your decisions.

(2 marks) Write out the regression equation and use it to predict the muscle mass of a 60-year-old woman.

(1 mark) How much greater muscle mass would you expect a woman age 45 to have compared to a 60-year-old woman?

(1 mark) What is the residual for a study subject of age 59 and muscle mass 70?

(1 mark) Notice that the plot of Standardized Residuals versus Age (bottom right plot) shows the presence of an outlier. Circle the outlier on the scatter plot (top left plot). Draw a new line on the scatter plot where you think the regression line would fall if the outlier were removed.

Call: lm(formula = muscle ~ age, data = muscle) Residuals -16.1804 -6.2469 -0.7458 6.7650 26.3045 Coefficients: Min 1Q Median 3Q Max Pr(>tl)

Explanation / Answer

1. It is an observational study but not an experiment because we are studying on a population which is not treated differently exclusively for our study.

2. The target population is the women aged greater than or equal to 40. The study population is the 60 women aged between 40-79 randomly selected from his hospital's database.

3. the differences might arise between target population and study population's results because the women selected for study population is from a hospital database and they are patients.Their muscle mass might behave differently than a normal woman.

4. The slope= -1.18341 , Intercept = 156.00172

5. The formula for SSR for a simple linear regression = (n-2)*(residual standard error)^2 = 58*8.329*8.329 = 4023.5899

r -squared = 1-(SSE/SST) = SSR/SST

therefore SST=SSR/r-squared = 4023.5899/0.7408 = 5431.41

SSE= SST-SSR

= 5431.41-4023.5899

=1407.8201

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