This is Business Statistics Exam.

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STAT 3155-EXAM 1 March 19, 2018

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1. (20pt) The following is a partial output when Y is regressed on X based on 20 observations ANOVA Table

Source df Sum of Squares Mean Square F

Regression ——- ——— ——- ——–

Error ——- 882.00 ——-

Total ——- 4410.00

Coefficients Table

Estimate Std. Error t value P(> |t|)

Intercept 3.432 ——- 0.265 0.7941

X ——- 0.142 -4.645 0.0001

n= r2 = r2a = s =

(a) (5pt) Complete the table above

(b) (5pt) Construct a 95% confidence interval for β1 and interpret your result.

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(c) (5pt) Test H0 : β1 = 0 against Ha : β1 6= 0 using α = 0.05.

(d) (5pt) What is the value of r, the correlation coefficient between X and Y ?

2. (40pt) A given company sells a special skin cream through fashion stores exclusively. It operates in 15 marketing districts and is interested in predicting district sales. To do so, it uses data on sales by district (Y) as well as district data on target population measured in thousands of persons (X1) and per capita discretionary income in dollars (X2). A partial regression output is given below.

df SS MS F

Regression — ——- —— ——– Residual — 56.88 —— Total —– 53901

Coefficients Table

Estimate Std. Error t value

Intercept 3.452 2.43 0.265

X1 0.496 0.006 ————

X2 0.009 0.0009 ————

(a) (5pt) Complete the ANOVA table above.

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(b) (5pt) Give the least squares regression equation and interpret the values b1 and b2, the estimates of β1 and β2, respectively.

(c) (5pt) Compute a 95% confidence interval for β1 and interpret your result

(d) (5pt) Test H0 : β2 = 0 against H1 : β2 6= 0. Use α = 0.05.

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(e) (5pt) Test H0 : β1 = β2 = 0 against H1 : at least one of them is not zero. Use α = 0.05.

(f) (2.5pt) Compute the coefficient of determination r2 for this model. Interpret you result.

(g) (2.5pt) Compute r2 adj

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(h) (5pt) What is the predicted sales by the model of a district with a population of 220,000 and a per capita discretionary income of $ 2500? If the Distance value of this district equals 0.50, construct a 95% prediction interval for its price. Interpret your result.

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(i) (5pt) Construct a 95% confidence interval for the average sales of district with a population of 220,000, a per capita discretionary income of $ 2500 and a Distance value of 0.50. Interpret your result.

3. (15pt) This problem relates to a construction project in which we wish to predict the profit (Y) using X1 (the size of the contract) and X2 (the supervisor’s experience) and we considering two competing models

Model I : Y = β0 + β1X1 + β2X2 + �

Model II : Y = β0 + β1X1 + β2X2 + β3X 2 1 + β4X

2 2 + �

If n = 20, SSE(Model I) = 100 and SSE(Model II) = 75.

(a) (5pt) Complete the following

df( SSE(Model I)) = −−−−−−−−−−−− df( SSE(Model II)) = −−−−−−−−−−−−

(b) (10pt) Test H0 : β3 = β4 = 0 Ha : Not H0.

Use α = 0.05. (Hint: Model II is obtained from Model I but setting β3 and β4 equal to zero)

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