Conducting hypothesis testing on multiple linear regression coefficients

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to conduct hypothesis testing on multiple linear regression coefficients.  Here's our problem statement: The coefficient beta-1 has a non-zero value that is helpful in predicting the value of the response variable.  If beta-1 is equal to zero, it is not helpful in predicting the value of the response variable and can be eliminated from the regression equation.  So to test the claim that beta-1 equals zero, use the test statistic t equals beta-1 minus zero divided by s-sub-b.  Critical values or P-values can be found using the t distribution with n minus k plus one degrees of freedom, where k is the number of predictor variables and n is the number of observations in the sample.  The standard error s is often provided by software.  For example, see the accompanying technology display which shows that s-sub-b-1equals 0.075033268, found in the column with the heading of standard error and the row corresponding to the first predictor variable of height.  Use the technology display to test the claim that beta-1 equals zero.  Also test the claim that beta-2 equals zero.  What do the results imply about the regression equation?

Part 1

OK, I can imagine a lot of students looking at this problem and thinking, What in the world have I got myself into here?  Because this sounds like a bunch of gobbledygook!  But really the problem is much more simple than what the problem statement would lead you to believe.  Because if we look here at our technology display, notice here how we've got the intercept, which has no coefficient, and then we've got two predictor variables here, height and weight --- excuse me, height and waist.  And we've got coefficient values for those variables here.  Standard error has been calculated here as was said in the problem statement.

So if we look at this first one, 0.075, we see that one is listed right here, surrounded in the red box.  But notice how all of this stuff here where we're talking about calculating a test statistic and P-values and all that stuff that we need for hypothesis testing, all of that's already calculated for us here in these columns here over at the end.  Notice this says "T stat" and this says "P-value."  So the values that we actually need are found here in this table.  And so we don't need to calculate anything, we just need to put the right numbers in the right places in our answer fields.

So to test the claim that beta1 equals zero, first we need to find our null hypothesis, which of course is a statement of equality.  Here our claim is a statement of equality.  Well, we can't have that be our alternative hypothesis because equality by definition belongs with the null hypothesis.  So this is going to be our null hypothesis.  So I select that option from the drop down.  The test statistic is located here in the table just as we got done talking about.  So beta-1 corresponds with the first predictor variable, which is height.  Intercept is not a predictor variable.  It's just an actual constant number there that's in your equation, so there's no variable associated with intercept.  The first predictor variable is height.  If I come over here and look at my t-statistic, my t-statistic is right here.  So all I have to do is put that number here in my answer field.  And I'm asked to round to three decimal places, so I'm going to do that here.

The P-value is right next door.  And we can see that with a P-value of zero, we're going to reject the null hypothesis, because with the P-value of zero, no matter what your — no matter what your critical values are, no matter what your confidence level is going to be, you know, the level of significance you're going to be inside that region of rejection.  And when you're inside the region of rejection, you reject the null hypothesis.  Well, if we're rejecting the null hypothesis, then that says that beta-1 is not equal to zero.  And so that means we're just going to keep the value that we see here listed in the table from our regression equation.  So I'm just going to put that value in here and say that it should be kept.  Well done!

Part 2

Now the next part wants me to test the claim that beta-2 equals zero.  Well, we're just going to go through the same process that we did before with the first hypothesis test.  So we're going to select a statement of equality for our null hypothesis.  The test statistic is located here on the last row because the last row corresponds with the second predictor variable, and that's where we're looking for beta-2 to correspond with the second predictor variable.  So here's our test statistic.  And we slip that in here, again rounding to three decimal places.

The P-value again is zero.  So that means we're inside the reason of rejection, we reject the null hypothesis, and that means we're just going to keep the same coefficient that we got here from the regression equation.  Fantastic!

Part 3

And now the last part of his problem asks, "What did the results imply about the regression equation?"  Well, we've kept both of the coefficients from our regression equation, so that tells us that, you know, we should include both of them in our regression equation.  So let's look through our answer options.  And I'm going to select the answer option that says we should include both of them.  And that's this one here.  Fantastic!

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