A psychologist conducted an experiment to investigate the relationship between w

Question

A psychologist conducted an experiment to investigate the relationship between worker productivity (Y, measured in units of thousands of machine castings), salary incentive, shift (day or night), and type of factory (unionized or non-unionized). The salary incentive was the bonus paid per worker, in dollars, for all castings produced in excess of 100 over a four week period. The multiple regression model is Y=0+1X1 + 2X2 + 3X3 + 4X1X2 + 5X1X3 + e, where X1 is the salary incentive, X2=1 for a dayshift worker and 0 otherwise, X3=1 if the worker is unionized and 0 otherwise. Data were collected for a random sample of 200 workers over a four week period. Four models were fit to the data.

The corresponding SSEs are below.

For which model is R2 the lowest?
1
2
3
4
[2 pt(s)]

Test to see whether interaction terms should be included in the model.What is the value of the test-statistic?  [3 pt(s)]

What are the degrees of freedom associated with the test-statistic? Numerator:  Denominator:  [1 pt(s)]

Select the interval below that contains the p-value for this test.
p-value 0.001
0.001 < p-value 0.01
0.01 < p-value 0.05
0.05 < p-value 0.1
0.1 < p-value 0.25
p-value > 0.25
[3 pt(s)]

Test to see if productivity differs between unionized and non-unionized workers.What is the value of the test-statistic?  [3 pt(s)]

What are the degrees of freedom associated with the test-statistic? Numerator:  Denominator:  [1 pt(s)]

Select the interval below that contains the p-value for this test.
p-value 0.001
0.001 < p-value 0.01
0.01 < p-value 0.05
0.05 < p-value 0.1
0.1 < p-value 0.25
p-value > 0.25

Model 1. Y=0+1X1 + 2X2 + 3X3 + 4X1X2 + 5X1X3 + e 2. Y=0+1X1 + 2X2 + 3X3 + e 3. Y=0+1X1 + 3X3 + e 4. Y=0+1X1 + 2X2 + e

Explanation / Answer

1.

SSE = SST (1-R2)

or, R2 = 1 - (SSE/SST)

So, R2 is lowest for which SSE is maximum.

So, R2 the lowest is lowest for model 4.

2.

The interaction term is in model 1. Let this be unrestricted model.

Model 2 has all other coefficients except the interaction term. Let this be restricted model.

F = [(SSER - SSEUR) / q] / [(SSEUR/(n-k)]

where SSER is SSE of the restricted model and SSEUR is SSE of unrestricted model.

q is number of coeffcients restricted. Here q = 2

n is number of samples. Here n = 200

k is number of coeffcients of unrestricted model (including intercept).

F = [(396 - 388) / 2] / [(388/(200-6)] = 2

Test statistic = 2

Degrees of freedom associated with the test-statistic - Numerator = q, Denominator = n - k

Numerator = 2, Denominator = 200-6 = 194

P-value for F = 2 and df = 2, 194 is 0.1381

So, the correct option is  0.1 < p-value 0.25

Test to see if productivity differs between unionized and non-unionized workers

The coeffcient is 3 for unionized and non-unionized workers

Let the model 2 be unrestricted model (with all X1, X2 and X3 variables)

Model 4 has all other coefficients except X3. Let this be restricted model.

F = [(SSER - SSEUR) / q] / [(SSEUR/(n-k)]

where SSER is SSE of the restricted model and SSEUR is SSE of unrestricted model.

q is number of coeffcients restricted. Here q = 1

n is number of samples. Here n = 200

k is number of coeffcients of unrestricted model (including intercept). k = 4

F = [(412 - 396) / 1] / [(396/(200-4)] = 7.92

Test statistic = 7.92

Degrees of freedom associated with the test-statistic - Numerator = q, Denominator = n - k

Numerator = 1, Denominator = 200-4 = 196

P-value for F = 7.92 and df = 1, 196 is 0.0054

So, the correct option is  0.001 < p-value 0.01

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