# Linear regression questions

A company has purchased several new, highly sophisticated machines. The production department needed some guidance with respect to qualifications needed by an operator. Is age a factor? Is the length of service as a machine operator important? In order to explore further the factors needed to estimate performance on the new machines, four variables are listed:

X1 = length of time employee was a machinist

X2 = Mechanical aptitude test score

X3 = prior on the job training

X4 = age in years

Performance of the new machine is designated Y.

Thirty machinists were selected at random. Data were collected for each and their performances on the new machines were recorded. The following regression equation was calculated:

Y^ = 11.6 + 0.4x1 + 0.286x2 + 0.112x3 + 0.002x4

(9.6) (0.772) (.75) (0.028) (0.0001)

NOTE: standard errors are in parenthesis.

1. Carl applied for a job on a new machine. He has been a machinist for six years and he scored 280 on the mechanical aptitude test. Carl's prior on-the-job performance rating is 97 and he is 35 years old. Estimate Carl's performance on the new machine.

2. Is mechanical aptitude test score significantly related to performance? Test using level of significance as 0.05. State the hypothesis to be tested, the decision rule, the test statistic and the decision.

3. Interpret the estimated coefficients on mechanical aptitude test score and prior on-the-job rating.

4. As age increases by one year, how much does estimated performance on the new machine increase, holding the other variables constant? Does the estimate seem reasonable?

5. It is argued by the machinists union that for each additional point increase in prior on-the-job rating, performance of the new machine will increase by only 0.05 units, and hence the prior on the job rating would not be an important determinate of future performance on a new machine. What does the data from the Company suggest about this statement? Level of significance is 0.05. State the hypothesis to be tested, the decision rule, the test statistic and the decision.

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I have provided detailed solutions to the 5 questions in the attached Word document.

A company has purchased several new, highly sophisticated machines. The production department needed some guidance with respect to qualifications needed by an operator. Is age a factor? Is the length of service as a machine operator important? In order to explore further the factors needed to estimate performance on the new machines, four variables are listed:

X1 = length of time employee was a machinist

X2 = Mechanical aptitude test score

X3 = prior on the job training

X4 = age in years

Performance of the new machine is designated Y.

Thirty machinists were selected at random. Data were collected for each and their performances on the new machines were recorded. The following regression equation was calculated:

Y^ = 11.6 + 0.4x1 + 0.286x2 + 0.112x3 + 0.002x4

(9.6) (0.772) (.75) (0.028) (0.0001)

NOTE: standard errors are in parenthesis.

1. Carl applied for a job on a new machine. He has been a machinist for six years and he scored 280 on the mechanical aptitude test. Carl's prior on-the-job performance rating is 97 and he is 35 years old. Estimate Carl's performance on the new machine.

Carl's estimated performance is calculated by substituting his 4 values into the regression equation. In Carl's case:

X1 = 6

X2 = 280

X3 = 97

X4 = 35

So,

Y^ = 11.6 + 0.4*(6) + 0.286*(280) + 0.112*(97) + 0.002*(35)

= 105.014

Carl's estimated performance is measure is 105.014.

2. Is mechanical aptitude test score significantly related to performance? Test using level of significance as 0.05. State the hypothesis to be tested, the decision rule, the test statistic and the decision.

To determine this you must test if the coefficient of X2 is significantly different from zero at the level. Since there are N = 30 participants in the experiment and p = 5 parameters estimated, the degrees of freedom for the t-test is N-p = 25. If we let denote the population coefficient X2 (our estimate of this from the sample is denoted ), then the test of hypothesis is as follows:

Hypotheses to be tested

Test Statistic:

Decision rule:

Reject the null hypothesis in favor of the alternative if the value of the test statistic is either greater than or less than .

From the t-tables with df = 25, = 2.06

Decision:

Since the test statistic was neither greater than 2.06, not less than -2.06, there is insufficient evidence to show that mechanical aptitude is significantly related to performance at the 0.05 level of significance.

3. Interpret the estimated coefficients on mechanical aptitude test score and prior on-the-job rating.

The estimated coefficient of mechanical aptitude in the regression equation is 0.286. You interpret this value by observing that if the other three variables in the equation are held constant the value of increases by 0.286 for each point increase in the mechanical aptitude score.

The estimated coefficient of prior on-job rating is 0.112. Similar to the coefficient of mechanical aptitude, this tells us that increases by 0.112 for each point increase in OJT rating when all other variables are held constant.

Note: From question 2, we have seen that the coefficient of mechanical aptitude is not significantly different from zero. Rather than keep it in the equation and interpret it as above, an experimenter in real life might want to remove it as a variable in the regression equation and refit using the other variables.

4. As age increases by one year, how much does estimated performance on the new machine increase, holding the other variables constant? Does the estimate seem reasonable?

As in question 3, this is the interpretation of the coefficient of X4. In this case the coefficient is 0.002, so the estimated performance of the new machine increases by 0.002 as age increases by one year. If we did a hypothesis test as in question 2, the test statistic would be 20, highly significant statistically. In this case the experimenter might want to ask if this result, even though statistically significant has a practical significance. Is this variable of practical significance since Y would increase by only one tenth of a point for a 50-year increase in age? One might conclude that even though age is significant statistically, it has very little or no practical significance.

5. It is argued by the machinists union that for each additional point increase in prior on-the-job rating, performance of the new machine will increase by only 0.05 units, and hence the prior on the job rating would not be an important determinate of future performance on a new machine. What does the data from the Company suggest about this statement? Level of significance is 0.05. State the hypothesis to be tested, the decision rule, the test statistic and the decision.

Here I assumed the company would want to show that the on-the-job rating will increase the performance by more than 0.05 units, so a one-tailed test was done. Had a two-tailed test been done, the decision rule would have been the same as in question 2, and the null would have still been rejected.

Hypotheses to be tested

Test Statistic:

Decision rule:

Reject the null hypothesis in favor of the alternative if the value of the test statistic is greater than .

From the t-tables with df = 25, = 1.71

Decision:

Since the test statistic was greater than 1.71, , there is sufficient evidence to show that that for each additional point increase in prior on-the-job rating, performance of the new machine will increase by greater 0.05 units (at the level of significance) and allow the company to argue that prior on the job rating would be an important determinate of future performance on a new machine.

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