1. Statistics 400 - Lecture 10

2. zLast day: 8.3 and started 8.4 zToday: Sections 8.4

3. Hypothesis Testing zHypothesis testing is a statistical technique to test if a conjecture about a population parameter is true zHas 4 Main Steps: yNull and Alternate Hypotheses yTest Statistic yP-Value yDecision based on pre-specified error rate

4. Example zHeights of one-year-old girls normally distributed with mean 30 inches and standard deviation of 1.2 inches zCompany claims taking 500 mg of Vitamin C makes the girls taller zIs the company’s claim true?

5. 1. Hypotheses zHypotheses are statements about a population and is expressed in terms of the population parameters zBegin by making an assumption of no change y(Treatment has no effect) zThis statement is called the null hypothesis (H 0 ) zTest will be designed to assess evidence against H 0

6. zHypothesis we suspect is true is called the alternate hypothesis (H 1 ) zAssume H 0 is true, collect data and see if there is evidence against H 0 and in favor of H 1

7. Example zHeights of one-year-old girls normally distributed with mean 30 inches and standard deviation of 1.2 inches zCompany claims taking 500 mg of Vitamin C makes the girls taller zH 0 : zH 1 :

8. 2. Test Statistic zTest statistic measures compatibility between H 0 and the data zIt is based on 2 principles: ybased on estimate of the parameter that appears in the hypotheses ymeasures distance of estimate from the hypothesized value zWhen H 0 is true, we expect the value of estimate to be close to parameter on average

9. Example (continued) zSuppose a random sample of 100 baby girls are given 500 mg of vitamin C daily for 1 year zMean height of the girls after 1 year is 32 inches (estimates population mean) zWhat is the distribution of if H 0 is true? zWhat is the distribution of if H 1 is true?

10. 3. P-Value zAssume that H 0 is true zThe P-value is the the probability of observing a test statistic as extreme or more extreme than the value actually observed when H 0 is true zWhat does a small p-value imply? zHow small is small?

11. Example (continued) zIf H 0 is true, the distribution of the sample mean is: zWhat does “extreme” mean in this context? zP-value=

12. 4. Decision zHow small must the p-value be to reject H 0 ? zMust decide which value of the test statistic give evidence in favor of H 1 zWould like the probability of observing such values to be small when H 0 is true zThe significance level of the test is:

13. Example (continued): zP-value= zSignificance level: zDecision:

14. Hypothesis Testing is Similar to a Jury Trial zH 0 : state of no change zH 1 : condition believed to be true zCollect data and compute test statistic zCompute p-value zReject or do not reject H 0 based on significance level z Not Guilty z Guilty z Collect evidence z Weigh evidence z Decide if evidence is in favor of guilty beyond a reasonable doubt

15. zHow do we interpret significance level zSome common significance levels: zHave we proven that H 0 is true or false?

16. Z-Test for the Population Mean zHave a random sample of size n ; x 1, x 2, …, x n z zTest Statistic: zCan be used for normal population or for large samples (why?)

17. Z-Test for the Population Mean (cont.) zP-value depends on the alternative hypothesis: y

18. Example: zScientists believe that abused children show elevated levels of depression zTo test this assertion, as random sample of 50 abused children were given a Profile of Moods States (POMS) test zThe results showed a mean depression score of 17.3 and standard deviation of 5.4 zTest, at the 5% level, whether abused children have a higher mean depression that that of the general population (mean=15)

19. Example: zA study titled “St. John’s Wort: Effect on CYP3A4 Activity” (Clinical Pharmacology and Therapeutics, 2000) reported a study that assesed urinary 6-beta-horoxycortisol/cortisol ratio in 12 subjects after 14 days of therapy with St. John’s Wort. zThe baseline mean ratio for the target population is 7.0 and the scientists wished to determine if the therapy resulted in increased a urinary 6-beta-horoxycortisol/cortisol ratio zUsing the data below, test this hypothesis