CK-12 Probability and Statistics - Advanced

9.3. Inferences about Regression http://www.ck12.org

regression model would be highly inaccurate. Therefore, when we estimate a linear regression model, we want to
ensure that the regression coefficient in the population(β)does not equal zero. Furthermore, it is beneficial to test
how strong (or far away) from zero the regression coefficient must be to strengthen our prediction of theYscores.

In hypothesis testing of linear regression models, the null hypothesis to be tested is that the regression coefficient
(β)equals zero. Our alternative hypothesis is that our regression coefficientdoes notequal zero.

H 0 :(β) = 0

Ha:(β) 6 = 0

We perform this hypothesis test similar to the previous conducted hypothesis test and need to next establish the
critical values for the hypothesis test. We use thet-distribution withn− 2 degrees of freedomto set such values. The
general formula used to calculate the test statistic for testing this null hypothesis is:

t=

observed value−hypothesized or predicted value

Standard Error of the statistic

=

b−β

sb

To calculate the test statistic for this regression coefficient, we also need to estimate the sampling distributions of the
regression coefficients. This statistic about this distribution that we will use is thestandard error of the regression
coefficient(sb)and is defined as:

Sb=

(

sy∗x

√

SSx

)

where:

sy∗x=the standard error of estimate

SSx=the sum of squares for the predictor variable(X)

Example:

Let’s say that the football coach is using the results from a short physical fitness test to predict the results of a longer,
more comprehensive one. He developed the regression equation ofY=. 635 X+ 1 .22 and the standard error of
estimatesY∗x=.56. The summary statistics are as follows:

Summarystatisticsfortwofootballfitnesstests.

n= (^24) ∑XY= 591. 50
∑X=^118 ∑Y=^104.^3
X ̄= 4. 92 Y ̄= 4. 35
∑X^2 =^704 ∑Y^2 =^510.^01
SSx= 123. 83 SSy= 56. 74
Using aα=.05, test the null hypothesis that, in the population, the regression coefficient is zero(H 0 :β= 0 ).
Solution:
We use thet-distribution for this test statistic and find that the critical values in thet-distribution at 22 degrees of
freedom(n− 2 )are 2.074 standard scores above and below the mean. Therefore,