Stat 112 Notes 3 Today: –Finish Chapter 3.3 (Hypothesis Testing). Homework 1 due next Thursday.

Confidence Intervals Point Estimate for slope: 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.

Property of Confidence Intervals

Computing Confidence Intervals I

Computing Confidence Interval II

Does Playing String Music Change Brain Activity? Studies have shown that activity can reorganize the human central nervous system. To study this further, psychologists used magnetic source imaging (MSI) to measure neuronal activity in the brains of nine string players (six violinists, two cellists and one guitarist) and six controls when the subject’s thumb and fifth finger of the left hand were exposed to mild stimulations. The psychologists felt that stringed instrument players, who use the fingers of their left hand extensively, might show different behavior in the brain – as a result of this extensive physical activity – than individuals who did not play stringed instruments. Specifically, they measured the amount of neuron activity in a region of the brain D5.

Simple Linear Regression Model

Hypothesis Testing Question

Hypothesis testing for slope Test statistic: Reject for (small/large, small, large) values of test statistic depending on. See Figure 3.15 in text book 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 For,p-value = Prob>|t| reported in JMP under parameter estimates. 2.One-sided test I: Reject if For 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 For,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.

Two Sided Test

One Sided Test (I)

One Sided Test (II)

One Sided Test of Non Zero Alternative

p-values for Poverty Rates and Doctors Regression

Example: One Sided Test

Example Continued: One and Two Sided Tests