PART 1

1. Consider the following data on the effect of a pesticide on mice. The basic response (y) was a measure of activity in the brain. It was postulated that brain activity would decrease with an increase in pesticide dosage, (x). This implies that the slope should be negative.

Dose (X), (mg/kg) 0.0 0.0 2.3 2.3 4.6 9.2 9.2 18.4 18.4 18.4

Activity (Y), (moles/liter/minute) 11.9 9.9 11.0 9.5 10.6 7.8 2.3 2.8 5.3 3.5

In addition to showing detailed step by step hand calculations, use SAS to check your work. Hand in hand calculations and the SAS program with output. Comment on the output wherever applicable. This is of high importance.

(a) Estimate the linear regression line. Give the estimated slope and intercept, and comment on your findings in a meaningful way. Also create a scatter plot of Y against X. Comment on it.

(b) Give a 90% confidence interval for the brain activity of a mouse receiving a pesticide with dose 6 mg/kg.

(c) Give a 95% confidence interval of the mean brain activity of mice that receive a dose of 17 mg/kg

(d) Test at alpha=0.05 whether the true slope is zero. Give the p-value.

(e) Give a 95% confidence interval for the true slope.

(f) Give a 99% confidence interval for the true intercept.

(g) Test at alpha=0.01 whether the true intercept is negative. Give the p-value.

(h) What percent of the total variability in brain activity is explained by the linear regression of brain activity on pesticide dose? Does you result show a good fit?

(i) Is the MSE much smaller than the sample variance for the response variable in this data set? Explain.

2. Consider an experiment on the weight loss (in mg), Y, of nine batches of Tribolium beetles after six days of starvation at nine different humidity levels (%), X.

X : 0 12 29.5 44 52 61 75.5 85.5 92.5

Y: 8.88 8.15 6.66 8.10 5.95 5.80 4.65 4.25 3.70

(a) Use simple linear regression to fit a regression line that allows you to predict weight from

humidity level. Comment on the resulting slope and intercept estimates.

(b) Give a 90% confidence interval for the true slope.

(c) Test at significance level 0.05 whether the true slope is non-negative vs. the alternative

hypothesis that the slope is negative. Comment on your findings.

(d) Estimate the weight loss of an individual beetle when subjected to a humidity level of 50%.

Use confidence level 95% .

(e) Calculate R squared. Comment on the result.

(f) Run the SAS program, and comment on the output.

(g) Create a scatter plot. Comment on it.

PART II

Use the data set on body fat with n=40;

1. Run proc reg with the full model:

proc reg ;

model body_fat = Hip Thigh Knee Ankle Biceps Forearm Wrist;

run;

Comment on what you can see in the SAS output on predicting body_fat from these variables.

2. run proc rsquare with options adjrsq mse cp. Select three "good models" in addition to the full model. Comment on how you chose these candidate models.

3. run prog reg to obtain the PRESS for each of these four models.

4. Create a table with a summary of your findings. Use as columns:

MODEL  ADJ RSQ      MSE     CP      PRESS

5. run proc stepwise to see what this algorithm would have given you as its "best model". Compare with your best model.

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Solution: The downloadable solution consists of 26 pages, 3166 words and 44 charts.
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