Hypothesis Testing

Method Deductive Reasoning –Reduce to 1 specific conclusion Either “reject” OR “don’t reject” the null hypothesis –From 4 ingredients that form the test: data from a random sample A “null” hypothesis (H 0 ) that will be the statistical focus of the procedure (≤, ≥, or =) An “alternative hypothesis (H 1 ) that is complement to the null (, or ≠ ) A “significance level” (  ) for the test.

The null hypothesis, H 0 Is an algebraic expression about a (population) parameter that is the focus of the testing. Must contain a limiting value, thus the algebraic statement MUST be an equality of some type: –H 0 :  = 400 “a bulb lives exactly 400 hours” –H 0 :  ≥ 21 “you must be at least 21 years old” –H 0 :  ≤ 60,000 “the tire cannot last more than 60,000 miles”

The alternative hypothesis The competing, mutually exclusive theory to the null hypothesis—the contratrian theory. It can NEVER be an algebraic expression with an equality … the H 1 MUST ALWAYS be an algebraic expression of a STRICT INEQUALITY –H 1 :  > 400 “the lightbulb lasts MORE THAN 400 hours, on average. –H 1 :  < 12% “fewer than 12% of voters expressed approval on the performance of US Legislators” –H 1 :  ≠ 2 “The 2-liter bottles filled at the softdrink plant do not contain 2 liters of soda.”

Page 285, 9.30 Question: Is  = 375 hours? Or Not?  = 100 hours … sigma is known (PhStat) Data: n=64, =350. Significance level: 5% –We are given a 5% chance at rejecting the null hypothesis even if the null is the Truth. H 0 :  = 375, versus H 1 :  ≠ 375 –We will reject the null hypothesis whenever the mean is NOT 375. So we can reject the null when the SAMPLE MEAN is way smaller or way larger than 375 … b/c  ≠ 375, means that the mean > or < 375.

Z Std error = 100/sqrt(64) = 12.5 hours  ≠ A 2-tail test of the null hypotheses: the test has two rejection regions for the null hypothesis B/C THE ALTERNATIVE HYPOTHESIS INVOLVES A: ≠ Conclusion: REJECT NULL, BECAUSE the data is 2 deviations from the theory, and I can be no farther than 1.96 deviations. Within reasonable doubt 360

July 17 P-value definition/interpretation Hypothesis testing for population proportions/percentages, . Start simple linear regression

Hypothesis testing, STATISTICAL VALUE approach DataNull, H0Alternative, H1 Significance level,  Test statistic: measures the actual difference between the evidence and the theory in the null hypothesis (in units of standard error Critical value or values: measures the tolerated difference that determines if the test statistic is “large enough” to reject the null hypothesis, or “insufficiently large” (don’t reject null) Conclusion about the null: binomial conclusion VERSUS

Hypothesis testing, PROBABILITY VALUE approach DataNull, H0Alternative, H1 Significance level,  Test statistic: measures the actual difference between the evidence and the theory in the null hypothesis (in units of standard error Conclusion about the null: binomial conclusion VERSUS P-VALUE: Probability value that measures the chance that the TEST STATISTIC indicates A WILLINGNESS TO ACCEPT THE NULL

Site example: statistic approach

Site example: p-value approach 1.We are allowing ourselves a 5% chance to make the error of rejecting the null hypothesis, even if it is true. 2.The p-value in this problem is This means that the evidence from the data indicates that we have a 9.62% chance that the null hypothesis is the TRUE ONE. 3.Since the chance that H0 is the true one > the chance that the H0 is not the true one, we must NOT REJECT THE H0.

Hypothesis testing: p-value approach