Checking the Assumptions of Simple Linear Regression Model Linearity and constant variance assumption generally look reasonable. Normality assumption looks reasonable except for one very negative residual Overall the simple linear regression model looks Reasonable.
Inference What is the relationship between X=cups of coffee drunk and Y=Time to bed for all Penn students? Inference: Drawing conclusions from a sample about a population. We can view our sample as a random sample from the population of Penn students (Are there any problems with this assumption?) Simple linear regression model Inference questions: –Confidence interval for slope – What is a plausible range of values for the true slope for Penn students? –Hypothesis testing: Is there evidence that cups of coffee drunk is associated with time to bed, i.e., is there evidence that ? Is there evidence that for each 1 additional cup of coffee drunk, the mean time to bed increases by at least half an hour, i.e., is there evidence that ?
Confidence Intervals Point Estimate: Confidence interval: range of plausible values for the true slope Confidence Interval: where is an estimate of the standard deviation of ( ) Typically we use a 95% CI. 95% CI is approximately 95% CIs for a parameter are usually approximately where the standard error of the point estimate is an estimate of the standard deviation of the point estimate.
Hypothesis testing for slope Test statistic: Reject for (small/large, small, large) values of test statistic depending on. See Figure 3.15 for the decision rules. p-value: Measure of how much evidence there is against the null hypothesis. Large p-values indicate no evidence against the null hypothesis, small p-values strong evidence against null. Generally accepted rule is to reject H_0 if p-value =0.05.
Risks of Hypothesis Testing Two types of errors are possible in hypothesis testing: –Type I error: Reject the null hypothesis when it is true –Type II error: Accept the null hypothesis when it is false. Probability of Type I error when H 0 is true = significance level of test, denoted by Probability of making correct decision when H a is true ( = 1-Prob. of Type II error) = power of test
Hypothesis Testing in the Courtroom Null hypothesis: The defendant is innocent Alternative hypothesis: The defendant is guilty The goal of the procedure is to determine whether there is enough evidence to conclude that the alternative hypothesis is true. The burden of proof is on the alternative hypothesis. Two types of errors: –Type I error: Reject null hypothesis when null hypothesis is true (convict an innocent defendant) –Type II error: Do not reject null hypothesis when null is false (fail to convict a guilty defendant)
Hypothesis Testing in Statistics Use test statistic that summarizes information about parameter in sample. Accept H 0 if the test statistic falls in a range of values that would be plausible if H 0 were true. Reject H 0 if the test statistic falls in a range of values that would be implausible if H 0 were true. Choose the rejection region so that the probability of rejecting H 0 if H 0 is true equals (most commonly 0.05) p-value: measured of evidence against H 0. Small p- values imply more evidence against H 0. p-value method for hypothesis tests: Reject H 0 if the p- value is. Do not reject H 0 if p-value is.
Scale of Evidence Provided by p-value p-valueEvidence against null hypothesis > 0.10No evidence 0.05 – 0.10Suggestive, but inconclusive 0.01 – 0.05Moderate < 0.01Convincing
Hypothesis Tests and Associated p-values 1.Two-sided test: Reject if p-value = Prob>|t| reported in JMP under parameter estimates. 2.One-sided test I: Reject if p-value = (Prob>|t|)/2 if t is negative 1-(Prob>|t|)/2 if t is positive
Hypothesis Tests and Associated p-values Continued 2.One-sided test II: Reject if p-value = (Prob>|t|)/2 if t is positive 1-(Prob>|t|)/2 if t is negative
Hypothesis Testing in JMP JMP output from Fit Line displays the point estimates of the intercept and slope, standard errors of the intercept and slope ( ), p-values from two-tailed tests of and.
Summary Chapter 3.3: We have developed methods of inference (confidence intervals, hypothesis tests) for the simple linear regression model. Important Note: These inferences are only correct if the simple linear regression model assumptions (linearity, constant variance, normality) are correct. It is important to check the assumptions. If the assumptions are approximately correct, then the inferences are approximately correct. Next class: Chapters 3.4-3.5: Assessing the fit of the regression line and prediction from the regression line.