Statistics 350
Sample Midterm 1
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1. Consider the simple linear regression model Yi= β0 + β1Xi + εi (i=1,2,…,n), where the εi’s are iid N(0,σ2). Suppose that β0=0.
a. What is the implication for the regression function?
b. How would the regression function plot on a graph?
c. Suppose that σ2 is known. State the likelihood function for a sample of size n.
d. Suppose that σ2 is known. Derive the maximum likelihood estimator for β1. (Note: show that it is a maximum)
2. Consider the simple linear regression model Yi= β0 + β1Xi + εi (i=1,2,…,n), where the εi’s are iid N(0,σ2). Suppose that β1=0.
a. Derive the least squares estimator for β0.
b. Show that your estimate for part (a) is a minimum.
c. Is your estimate in part (a) an unbiased estimator? Prove or disprove that it is unbiased for β0.
d. Calculate ∑ei for this model. What does this imply about the residuals and a plot of residuals versus X?
3. A substance used in biological and medical research is shipped by air-freight to users in cartons of 1,000 ampules. The data below, involving 10 shipments were collected on the number of times the carton was transferred from one aircraft to another over the shipment route (X) and the number of ampules found to be broken upon arrival (Y). Assume the simple linear regression model is appropriate.
X / Y1 / 16
0 / 9
2 / 17
0 / 12
3 / 22
1 / 13
0 / 8
1 / 15
2 / 19
0 / 11
Some summary statistics for these data are:
A simple linear regression model was fit to these data. The computer output is summarized below.
a. Test the hypothesis that there is a linear association between the number of transfers for a shipment and the number of broken ampules. State the hypotheses, value of your test statistic and your conclusion.
b. Estimate the expected number of broken ampules when there are 3 transfers.
c. Give a 95% confidence interval for the expected number of broken ampules when there are 3 transfers for the shipment.
d. A customer for the shipping company claims that, on average, there are more than 20 broken ampules when there are 3 transfers for the shipments. Test this hypothesis at a 5% significance level.
e. A plot of the residuals versus the number of transfers is shown below. From the plot, is there any evidence that the assumptions of the simple linear regression model are violated (explain your answer)?
4.