WGU Influence of Calcium on The Contraction of Heart Muscle in Rats Question You will need R only for Question 1.Please submit your R code as well as the p

WGU Influence of Calcium on The Contraction of Heart Muscle in Rats Question You will need R only for Question 1.Please submit your R code as well as the pdf with all your solutions. Midterm 1 (100 pts)
Name & Class:
Instructions
• You have one-hour and fifteen minutes.
Question 1 (60 points):
(Use R for data analysis)
The purpose of this experiment was to assess the influence of calcium in solution on the contraction of heart
muscle in rats. The left auricle of 21 rat hearts was isolated and on several occasions a constant-length strip
of tissue was electrically stimulated and dipped into various concentrations of calcium chloride solution, after
which the shortening of the strip was accurately measured as the response.
The data are stored in R package MASS. You can look at them with commands attach(muscle), names(muscle),
summary(muscle), muscle. A linear regression model is used to predict the change in length of the strip
(variable Length, in mm) based on the concentration of calcium chloride solution (variable Conc, in multiples
of 2.2 mM).
(a) Calculate the equation of the sample regression line that predicts Length based on Cone.
(b) Complete the ANOVA table and estimate the variance of Length.
(c) Compute a 95% confidence interval for the regression slope β1 .
(d) Test whether the slope is zero or not.
(e) Calculate the percent of total variation explained by this regression model.
(f) Compute a 90% confidence interval for the mean Length when the concentration of calcium is 2.5.
(g) Compute a 90% prediction interval for Length if the concentration of calcium is 2.5.
(h) Verify the standard regression assumptions-normality and homoscedasticity. Report p-values and state
your conclusions.
(i) (Graduate only) Find the optimal Box-Cox transformation. Does it improve normality of residuals?
(j) (Graduate only) Test the model for the lack of fit.
#Answer:
1
2
#
Question 2 (30 points):
Linear regression was used to model the relationship between X, the protein intake (in grams of protein per
kilogram of body weight per day) and Y , the nitrogen balance (in milligrams of nitrogen per kilogram of
body weight per day). Results of tests on n = 5 patients are summarized in the table. The sample correlation
coefficient is equal to 0.9. Hint: For this problem you may use that SSR = r2 SST and R2 = SSR
SST .
X, protein intake
Y , nitrogen balance
Sample mean
0.79
20.6
Standard deviation
0.24
45.4
(a) Calculate the equation of the sample regression line that predicts Y based on X.
(b) Complete the ANOVA table. Include sums of squares, degrees of freedom, mean squares, and the
ANOVA F -statistic.
(c) What percent of the total variation of nitrogen balance is explained by the protein intake?
(d) For α = 0.05 level, is the linear relation between X and Y found significant? Conduct the F -test and
state your conclusion. The table of F -distribution is given.
(e) Estimate Var(Y ), the population variance of nitrogen balance.
#Answer:
3
#
Question 3 (10 points):
Refer to regression model Yi = β0 + β1 Xi + i . How would the regression function plot on a graph if β1 = 0?
Derive the least squares estimator of β0 for this model.
#Answer:
#
4

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