Unformatted text preview: nse we had to estimate 2
quantities, the y-intercept and the slope; so we lose 2 df. Estimating the Standard Deviation: Exam 2 vs Final Below are the portions of the SPSS regression output that we could use to obtain the estimate of for our regression analysis. Model Summaryb
.889a R Square
.738 Std. Error of
8.24671 a. Predictors: (Constant), exam 2 scores (out of 75)
b. Dependent Variable: final exam scores (out of 100) ANOVAb
Total Sum of
5 Mean Square
68.008 a. Predictors: (Constant), exam 2 scores (out of 75)
b. Dependent Variable: final exam scores (out of 100) 194 F
.018a Significant Linear Relationship? Consider the following hypotheses: H 0 : 1 0 versus H a : 1 0 What happens if the null hypothesis is true? If 1=0 then E(Y) = 0 => a constant no matter what the value of x is.
i.e. knowing x does not help to predict the response. So these
hypotheses are testing if there is a significant non-zero linear
relationship between y and x.
There are a number of ways to test this hypothesis. One way is through a t‐test statistic (think about why it is a t and not a z test). sample statistic - null value The general form for a t test statistic is: t standard error of the sample statistic We have our sample estimate for 1 , it is b1 . And we have the null value of 0. So we need the standard error for b1 . We could “derive” it, using the idea of sampling distributions (think about the population of all possible b1 values if we were to repeat this procedure over and over many times). Here is the result: t‐test for the population slope 1 b 0
To test H 0 : 1 0 we would use t 1 s.e.(b1 )
s where SE (b1 ) x x 2 and the degrees of freedom for the t‐distribution are n – 2. This t‐statistic could be modified to test a variety of hypotheses about the population slope (different null values and various directions of extreme). Try It! Significant Relationship between Exam 2 Scores and Final Scores?...
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