Unformatted text preview: Is there a significant (non‐zero) linear relationship between the exam 2 scores and the final exam scores? (i.e., is exam 2 score a useful linear predictor for final exam score?) That is, test H 0 : 1 0 versus H a : 1 0 using a 5% level of significance. 1. SE (b1 ) 2. t s x x 2 8.24671 0.269 940 b1 0 1.046 0 3.89
s.e.(b1 )
0.269 3. Using table A.3 with df = 6 – 2 = 4, we have pvalue < 2(0.020) = 0.04
We can reject H0 and conclude the exam 2 score is a significant linear predictor of
final exam score. 195 Think about it: Based on the results of the previous t‐test conducted at the 5% significance level, do you think a 95% confidence interval for the true slope 1 would contain the value of 0? Confidence Interval for the population slope 1 b1 t * SE b1 where df = n‐2 for the t * value Compute the interval and check your answer. Could you interpret the 95% confidence level here? 1.046 ± (2.78)(0.269) 1.046 ± 0.748 ( 0.298, 1.794 )
(t* = 2.78 from df = 4 and 95% confidence) If this experiment were repeated many times, we’d expect 95% of the
resulting confidence intervals to contain the population slope 1. Inference about the Population Slope using SPSS Below are the portions of the SPSS regression output that we could use to perform the t‐test and obtain the confidence interval for the population slope 1 . Coefficientsa Model
1 (Constant)
exam 2 scores (out of 75) Unstandardized
Coefficients
B
Std. Error
21.667
14.125
1.046
.269 Standardized
Coefficients
Beta
.889 t
1.534
3.888 Sig.
.200
.018 a. Dependent Variable: final exam scores (out of 100) Note: There is a third way to test H 0 : 1 0 versus H a : 1 0 . It involves another F‐test from an ANOVA for regression. ANOVAb
Model
1 Regression
Residual
Total Sum of
Squares
1027.967
272.033
1300.000 df
1
4
5 Mean Square
1027.967
68.008 F
15.115 Sig.
.018a a. Predictors: (Constant), exam 2 scores (out of 75)
b. Dependent Variable: final exam scores (out of 100) * The ttest is more flexible than the F test; F only twosided with null=0 196 Predicting for Individuals versus Estimating...
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 Summer '10
 Gunderson
 Statistics, Correlation

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