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 pvalue 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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