Quality Quiz from Professor Cleary

Congratulations:
"B" is correct.

Click here for a more complete video explanation

The t value has nothing to do with tons of coal mined, but is instead a statistical term that is used to test the significance of b1, the regression coefficient for X1, tons of coal mined.

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 and 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.

Let's start with tons mined. To get some insight on this question, we'll recall a figure from the November 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 indeed affect the cost.

The question is whether one can say with statistical accuracy that tons of coal really do affect the cost of mining coal.

In testing for independent variables for multiple regression, begin by establishing the null, that the independent variables (X1 and X2) are not a predictor of the dependent variable (Y). Then proceed to accept or reject that hypothesis. If the null that the independent variables (X1 or X2) do not represent a good predictor of Y is rejected, one accepts the opposite that X1 or X2 are predictors of Y—something that may seem slightly counterintuitive. Let's start with X1 tons of coal.

For this case:

Step 1

Interpretation: The null hypothesis is that X1, amount of coal mined, does 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 t value to test this hypothesis:
Data:

Interpretation: We have assumed that X and Y are not related. Therefore , the hypothetical regression coefficient, must be equal to .0 The calculated value for b1, which is an estimate for , is equal to .4. The amount of error that occurred in calculating b1 is .1265.

The appropriately calculated t value is 3.162. The larger the calculated t, 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 X and Y.

Step 4

Compare the calculated t to the tabular t, to determine whether to accept or reject the null hypothesis. In this case, the tabular t is 2.4469 and the calculated is t 3.162.

Since the calculated t is greater than the tabular t, we must reject that =0, and therefore conclude that X1 does indeed have an effect on the value of Y.

See the video to learn what the p-value of .007 for tons mined (in the printout) actually means.

Polly knew the significance of the t statistic, bringing her reputation for accuracy to a higher level than Kohl Minor ever dreamed of attaining—and all because of her SPC training background.

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