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 oppositesomething 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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