1. STT 315 Ashwini Maurya Acknowledgement: Author is indebted to Dr. Ashok Sinha, Dr. Jennifer Kaplan and Dr. Parthanil Roy for allowing him to use/edit many of their slides.

2. Review of last lecture … Started with confidence interval and Testing of hypothesis; i) Confidence interval is used for obtaining a range ( interval) of true population parameter value with certain confidence. ii) One sample “Testing of hypothesis” is useful in testing a claim or an Hypothesis based on sample data. Goals of this chapter: To extend the idea for two samples. Write and interpret a confidence interval for the difference between two proportions. Perform a hypothesis test comparing two proportions and interpret the results.

3. Chapter 22 Using inference to compare two groups. An example: Is the percentage of undergraduate MSU women who smoke the same as the percent of undergraduate MSU men who smoke? Here the objective is to i)find the confidence interval of “difference of proportions for male and female smokers”, and ii)Testing the hypothesis if two proportions are same or differ significantly.

4. Things that are the same A confidence interval is still of the form: Statistic ± M.E. We are still C% confident that the true parameter value lies in the calculated C% confidence interval. A hypothesis test still consists of: – Writing hypotheses ( Null H0 and Alternative Ha.) – Checking conditions – Finding a p-value ( also called P( Type I error)) – Writing a conclusion in the context of the problem. ( Making a decision either in favor of Null or Alternative Hypothesis)

5. Conditions Independence – Randomization or Random Sample – 10% condition – (we will add one more soon) Sample size : both groups must have at least 10 successes and 10 failures.

6. Example 1 Research question: Is the proportion of female undergraduate MSU students who smoke the same as the proportion of male undergraduate MSU students who smoke Step 1: Write the hypotheses H 0 : p m = p f against H a : p m ≠ p f. A sample of ??????????????

7. Which of the following is the Type I error for this test? A.We conclude that the rate of smoking is the same for college aged men and women but it actually is not. B.We conclude that the rate of smoking is not the same for college aged men and women but it actually is. C.The probability that our test will find a difference between the rates of smoking for college aged men and women if a difference actually does exist. D.The size of the difference in the rates of smoking in college aged men and women if one exists.

8. Have you smoked a tobacco cigarette in the last week? A.Yes and I am a female B.Yes and I am a male C.No and I am a female D.No and I am a male 15.6% of men in the sample smoked 13.6% of women in the sample smoked In the sample, 2% more men than women smoked.

9. Conditions for inference with two proportions Independence – Randomization or random sample – 10% condition – Samples from the two populations are independent Sample size : both groups must have at least 10 successes and 10 failures

10. Conditions for inference with two proportions Independence - in the previous cigarette example we discussed the issues with independence – Random Sample: met – 10% condition: met – Samples from the two populations are independent: probably met Sample size : both groups must have at least 10 successes and 10 failures Is the sample size condition met by our data? A. YesB. No - only 5 men and 8 women smoke and both are less than 10

11. Step 3 - Describe and Draw the distribution of the statistic What is the statistic? What is its distribution?

12. Theory behind inference comparing two proportions Based on a results, which state that for independent random variables means and variances add, since we know that sampling distribution of a proportion has a normal shape with mean, p, and standard deviation,, we derive that the sampling distribution for the parameter also has a normal shape with mean, p 1 - p 2, and standard deviation,

13. 1.Specify the hypothesis being tested - H 0 : p m = p f - H a : p m ≠ p f 2.Check Conditions –Not a random sample, but less than 10% of the population. –The men and women in the class are probably independent –There are only 5 male smokers and 8 female. Both are less than 10 so the sample is not large enough 3.### Draw the expected distribution of the sample statistic #### –N(0, ) 4.Calculate the p-value - Using 2propZtest –z = 0.27, p-value = 0.788 5.Write the conclusion –If the percent of male and female undergraduates at MSU who smoke are the same, we would have seen a 2% or more difference in about 78.8% of samples of this size. This large p-value provides no evidence to reject the null hypothesis. We conclude that there is no difference in the percent of men and women at MSU who smoke.

14. If we have made an error, what kind of error did we make? A.Type I B.Type II

15. Two Proportion Hypothesis Tests Procedure - Summary Add the condition that the two samples be independent of each other Null hypothesis: H 0 : p 1 = p 2 (OR H 0 : p 1 - p 2 = 0). Use pooled proportion, p pooled, in calculation of SD (the assumption is that they are equal) Formula for SE =

16. Two Proportion Confidence Interval Procedure - Summary Add the condition that the two samples be independent of each other Statistic is: Formula for SE = Make a 95% confidence interval for the difference in the percent of male students and female students who smoke.