Quality Quiz from Professor Cleary

Congratulations:
"A" is correct.

Click here for a more complete video explanation

Polly Yurathane is batting a thousand with this response. As we know from last month's problem, in multiple regression, there are two questions to ask about the validity of the regression equation that is derived. First, whether the individual independent variables are truly predictors of Y (cost): In this case, the question would be whether the tons of coal mined or depth of the shaft are truly drivers for the cost of coal mined. The second question is whether the tons of coal mined and depth of shafts considered as a pair have an effect on the cost of coal mined.

Last month we looked at individual independent variables, using the t statistic. This month, we will examine all the independent variables as a group. In this case, this means both X1 and X2. To get some insight on this question, we’ll recall a figure from the December Quality Quiz, showing the same data. The calculations are as follows:

(the regression equation)

The balloons represent the equation’s 9 data points that fit the plan well. It appears that the tons of coal mined and depth of shaft both indeed affect the cost of mining coal.

The question is whether one can say with statistical accuracy that this is true.

Once again, we can turn to hypothesis testing to solve the problem.

In testing for independent variables for multiple regression, begin by establishing the null, that the independent variables (X1, X2) are not a predictor of the dependent variable (Y). Then proceed to accept or reject that hypothesis. If one rejects the null that the independent variables (X1, X2) do not represent a good predictor of Y, one accepts the opposite--something that may seem slightly counterintuitive.

For this case:

Step 1

Interpretation: The null hypothesis is that X1, amount of coal mined, and X2, the depth of the mine shaft, do not affect Y, the cost of mining coal. The alternative is the opposite.

Step 2

Interpretation: We are willing to accept a 5% chance of committing a Type I error (rejecting the null when it is in fact true).

Step 3

Calculate the appropriate F value to test this hypothesis:
Data: F = 2.44172E+32

Interpretation: We have assumed that X1 and X2 are not related. Therefore and , the hypothetical regression coefficients, must be equal to .0

The appropriately calculated F value is 2.44172E+32 (Note: "E+32" indicates that the decimal is placed 32 places to the right, creating a very large number.)

The larger the calculated F, the more likely it is that we will reject the null hypothesis , = 0. If the null is rejected, the conclusion is that there is a relationship between X1, X2, and the dependent variable Y.

Step 4

Compare the calculated F to the tabular F, to determine whether to accept or reject the null hypothesis. In this case, the tabular F is 5.14 and the calculated F is 2.44172E+32.

Since the calculated F is greater than the tabular F, we must reject the null, and therefore conclude that X1 and X2 are predictors of Y.

Let's hope Polly has kept Hyde happy with her knowledge.

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