Statistics 701

Homework #1

Date Assigned: 1/24/00

Date Due: 1/31/00

  1. (Problem 1.27 on page 40 of the textbook) A person's muscle mass is expected to decrease with age. To explore this relationship in women, a nutritionist randomly selected four women from each 10-year age group, beginning with age 40 and ending with age 79. The resulting data set is given below, where X = Age and Y = measure of muscle mass. Assume that a first order regression model is appropriate.

Y= muscle mass / X = Age
82.0 / 71.0
91.0 / 64.0
100.0 / 43.0
68.0 / 67.0
87.0 / 56.0
73.0 / 73.0
78.0 / 68.0
80.0 / 56.0
65.0 / 76.0
84.0 / 65.0
116.0 / 45.0
76.0 / 58.0
97.0 / 45.0
100.0 / 53.0
105.0 / 49.0
77.0 / 78.0

For the following problems, do the computations by hand instead of using Minitab or SAS. Or you may use Excel to do the computations using the spreadsheet capabilities of Excel, not its statistical functions. This way you will get a feel of how the calculations proceed.

a)Present a scatterplot of this data set (you may generate this using Minitab). Examining this data set, assess whether a first-order linear regression model is viable.

b)Compute the sample correlation coefficient between X and Y.

c)Obtain estimates of 0 and 1, the y-intercept and slope of the regression line. Provide interpretations of these two estimates.

d)Compute the residual at X = 65 years old.

e)Obtain an estimate of 2, the common variance of the error terms.

f)Obtain estimates of the standard errors of the estimates of 0 and 1.

g)Obtain the coefficient of determination, R2, of the fitted model. Is this R2 equal to the square of the sample correlation coefficient? Is this coefficient of determination large so as to be able to conclude that the fitted model is good?

h)Test the hypothesis that 1 is 0 versus the hypothesis that it is not zero. Use a 5% level of significance. Equivalently, construct a 95% confidence interval for 1.

i)Test the hypothesis that 0 is 0 versus the hypothesis that it is not zero. Use a 5% level of significance. Equivalently, construct a 95% confidence interval for 0. Would it have been more plausible to fit a model where 0 = 0, and if so, why?

j)Obtain an estimate of the mean muscle mass for a woman of age 60 years old. Provide an estimate of the standard error for your estimate, and construct a 95% confidence interval for the mean muscle mass for a woman of age 60 years old.

k)Construct a 95% prediction interval for the value of the muscle mass of a woman of age 60 years old. How does this prediction interval compare with the confidence interval you obtained in (g)?

  1. Using the data above, use Minitab or SAS to fit a simple linear regression model. Also, generate the confidence and prediction bands and plot the fitted regression line to the scatterplot of the data (best to use Minitab for this using the "Fitted Line Plot"). Are the results you obtained in part (1) consistent with the values you obtained using Minitab or SAS?
  1. The BodyFat data set can be obtained from our website in the Lectures Folder. The Minitab File is "bodyfatproject.mpj". Using Minitab, do the following:

a)Fit simple linear regressions for PerBodyFat (percent body fat) versus Age, with Age as predictor variable.

b)Fit simple linear regressions for PerBodyFat versus AbdoCirc (abdominal circumference), with AbdoCirc as predictor variable.

c)Fit simple linear regressions for PerBodyFat versus NeckCirc (neck circumference), with NeckCirc as predictor variable.

d)Which of the three models will be best for predicting PerBodyFat? Explain your choice.

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