1. AP Statistics Chapter 11 Section 2

2. TestConfidence IntervalFormulasAssumptions 1-sample z-test mean SRS Normal pop. Or large n (n>40) Know 1-sample t-test mean (could also be used for a matched pairs design) SRS < 15 -- very normal 15 < n < 40 -- can be used except in the presence of outliers or strong skewness > 40 -- use regardless of normality 2-sample z-test SRSs Normal pops or n>40, sigma 2-sample t-test SRS < 15 -- very normal 15 < < 40 can be used except in the presence of outliers or strong skewness > 40 -- use regardless of normality 1-proportion TESTConfidence IntervalFormula(s)Assumptions 2-proportion Goodness of fit 1-way table

3. Two-Sample Problems The goal of inference is to compare the responses to two treatments or to compare the characteristics of two populations. We have a separate sample from each treatment or each population

4. Examples A medical researcher is interested in the effect on blood pressure of added calcium in our diet. She conducts a randomized comparative experiment in which one group of subjects receives a calcium supplement and a control group gets a placebo. A psychologist develops a test that measures social insight. He compares the social insight of male college students with that of female college students by giving the test to a large group of students of each gender.

5. Assumptions Two SRSs from two distinct populations. The samples are independent. We measure the same variable for both samples. Both populations are normally distributed. The means and standard deviations of the populations are unknown.

6. Does increasing the amount of calcium in our diet reduce blood pressure? An observational study of a large sample of people revealed a relationship between calcium intake and blood pressure. The relationship was strongest for black men. An observational study does not establish causation – A randomized comparative experiment is designed to determine the relationship.

8. Treatment Group Data (G1)Control Group Data (G2) (original BP – BP after calcium supplement) (original BP – BP after placebo) 7 -412 18 17-3 3 -5 15 102 11-11 -2 -3

9. Remember: A negative sign indicates an increase in BP after the “treatment”.

10. Is this outcome good evidence that calcium decreases BP in the entire population of healthy black men more than a placebo does? RUN THE TEST Treatment group Control group

11. Assumptions: 1.Two SRSs 2.Normal ? Treatment Group Control Group

12. Normal? The control group’s distribution seems roughly normal. The Treatment group’s data has an irregular distribution – could be due to the small sample size. The irregularity is not extreme enough to prevent the use of t- procedures. Assumptions met.

13. Degrees of Freedom rule for two-sample problems. Use the smaller of the two sample sizes OR use calculator. The formula for finding the d of f of two-sample problems can be found on pg. 633. (You will not need to show use of the formula for the AP Exam)

14. T(15.59) = 1.604 P-value =.0644 Given that the calcium did not reduce BP the observed difference would occur approx. 6 out of every 100 times just by chance therefore this evidence supports the statement that the calcium did not make a difference in BP.

15. 95% CI

16. Pooling The two-sample t-test is valid whether or not the population variances are equal. A choice for you is to assume that the two population variances are equal – this procedure averages the two variances to estimate the common population variance. This is called the pooled two-sample test statistic. Not used often.