1. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 1 of 27 Chapter 11 Section 3 Inference about Two Population Proportions

2. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 2 of 27 Chapter 11 – Section 3 ●Learning objectives  Test hypotheses regarding two population proportions  Construct and interpret confidence intervals for the difference between two population proportions  Determine the sample size necessary for estimating the difference between two population proportions within a specified margin of error 2 1 3

3. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 3 of 27 Chapter 11 – Section 3 ●Learning objectives  Test hypotheses regarding two population proportions  Construct and interpret confidence intervals for the difference between two population proportions  Determine the sample size necessary for estimating the difference between two population proportions within a specified margin of error 2 1 3

4. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 4 of 27 Chapter 11 – Section 3 ●This progression should not be a surprise ●One mean and one proportion  Chapter 9 – confidence intervals  Chapter 10 – hypothesis tests ●Two means  Sections 11.1 and 11.2 – hypothesis tests and confidence intervals ●Now for two proportions …

5. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 5 of 27 Chapter 11 – Section 3 ●We now compare two proportions, testing whether they are the same or not ●Examples  The proportion of women (population one) who have a certain trait versus the proportion of men (population two) who have that same trait  The proportion of white sheep (population one) who have a certain characteristic versus the proportion of black sheep (population two) who have that same characteristic

6. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 6 of 27 Chapter 11 – Section 3 ●The test of two populations proportions is very similar, in process, to the test of one population proportion and the test of two population means ●The only major difference is that a different test statistic is used ●The test of two populations proportions is very similar, in process, to the test of one population proportion and the test of two population means ●The only major difference is that a different test statistic is used ●We will discuss the new test statistic through an analogy with the hypothesis test of one proportion

7. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 7 of 27 Chapter 11 – Section 3 ●For the test of one proportion, we had the variables of  The hypothesized population proportion (p 0 )  The sample size (n)  The number with the certain characteristic (x)  The sample proportion ( ) ●For the test of one proportion, we had the variables of  The hypothesized population proportion (p 0 )  The sample size (n)  The number with the certain characteristic (x)  The sample proportion ( ) ●We expect that should be close to p 0

8. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 8 of 27 Chapter 11 – Section 3 ●In the test of two proportions, we have two values for each variable – one for each of the two samples  The two hypothesized proportions (p 1 and p 2 )  The two sample sizes (n 1 and n 2 )  The two numbers with the certain characteristic (x 1 and x 2 )  The two sample proportions ( and ) ●In the test of two proportions, we have two values for each variable – one for each of the two samples  The two hypothesized proportions (p 1 and p 2 )  The two sample sizes (n 1 and n 2 )  The two numbers with the certain characteristic (x 1 and x 2 )  The two sample proportions ( and ) ●We expect that should be close to p 1 – p 2

9. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 9 of 27 Chapter 11 – Section 3 ●For the test of one proportion, to measure the deviation from the null hypothesis, we took which has a standard deviation of

10. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 10 of 27 Chapter 11 – Section 3 ●For the test of two proportions, to measure the deviation from the null hypothesis, it is logical to take which has a standard deviation of

11. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 11 of 27 Chapter 11 – Section 3 ●For the test of one proportion, under certain appropriate conditions, the difference is approximately normal with mean 0, and the test statistic has an approximate standard normal distribution

12. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 12 of 27 Chapter 11 – Section 3 ●Thus for the test of two proportions, under certain appropriate conditions, the difference is approximately normal with mean 0, and the test statistic has an approximate standard normal distribution

13. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 13 of 27 Chapter 11 – Section 3 ●For the particular case where we believe that the two population proportions are equal, or p 1 = p 2 and

14. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 14 of 27 Chapter 11 – Section 3 ●Now for the overall structure of the test  Set up the hypotheses ●Now for the overall structure of the test  Set up the hypotheses  Select the level of significance α ●Now for the overall structure of the test  Set up the hypotheses  Select the level of significance α  Compute the test statistic ●Now for the overall structure of the test  Set up the hypotheses  Select the level of significance α  Compute the test statistic  Compare the test statistic with the appropriate critical values ●Now for the overall structure of the test  Set up the hypotheses  Select the level of significance α  Compute the test statistic  Compare the test statistic with the appropriate critical values  Reach a do not reject or reject the null hypothesis conclusion

15. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 15 of 27 Chapter 11 – Section 3 ●In order for this method to be used, the data must meet certain conditions  Both samples are obtained independently using simple random sampling ●In order for this method to be used, the data must meet certain conditions  Both samples are obtained independently using simple random sampling  The number of successes and the number of failures for each sample are greater than or equal to 10 ●In order for this method to be used, the data must meet certain conditions  Both samples are obtained independently using simple random sampling  The number of successes and the number of failures for each sample are greater than or equal to 10  Each sample size is no more than 5% of the population size ●In order for this method to be used, the data must meet certain conditions  Both samples are obtained independently using simple random sampling  The number of successes and the number of failures for each sample are greater than or equal to 10  Each sample size is no more than 5% of the population size ●These are the usual conditions we need to make our test of proportions calculations

16. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 16 of 27 Chapter 11 – Section 3 ●State our two-tailed, left-tailed, or right-tailed hypotheses ●State our level of significance α, often 0.10, 0.05, or 0.01

17. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 17 of 27 Chapter 11 – Section 3 ●Compute the test statistic which has an approximate standard normal distribution ●Compute the critical values (for the two-tailed, left-tailed, or right-tailed test)

18. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 18 of 27 Chapter 11 – Section 3 ●Each of the types of tests can be solved using either the classical or the P-value approach ●Based on either of these two methods, do not reject the null hypothesis

19. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 19 of 27 Chapter 11 – Section 3 ●We have two independent samples  55 out of a random sample of 100 students at one university are commuters  80 out of a random sample of 200 students at another university are commuters  We wish to know of these two proportions are equal  We use a level of significance α =.05 ●We have two independent samples  55 out of a random sample of 100 students at one university are commuters  80 out of a random sample of 200 students at another university are commuters  We wish to know of these two proportions are equal  We use a level of significance α =.05 ●When we calculate np(1-p) for each of the two samples, we get values of 24.75 and 48, so our method can be used

20. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 20 of 27 Chapter 11 – Section 3 ●The test statistic is ●The critical values for a two-tailed test using the normal distribution are ± 1.96, thus we reject the null hypothesis ●The test statistic is ●The critical values for a two-tailed test using the normal distribution are ± 1.96, thus we reject the null hypothesis ●We conclude that the two proportions are significantly different

21. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 21 of 27 Chapter 11 – Section 3 ●Learning objectives  Test hypotheses regarding two population proportions  Construct and interpret confidence intervals for the difference between two population proportions  Determine the sample size necessary for estimating the difference between two population proportions within a specified margin of error 2 1 3

22. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 22 of 27 Chapter 11 – Section 3 ●Confidence intervals are of the form Point estimate ± margin of error ●Confidence intervals are of the form Point estimate ± margin of error ●We can compare our confidence interval with the test statistic from our hypothesis test  The point estimate is  We use the denominator of the test statistic as the standard error  We use critical values from the normal distribution

23. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 23 of 27 Chapter 11 – Section 3 ●Thus confidence intervals are Point estimate ± margin of error Standard error Point estimate

24. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 24 of 27 Chapter 11 – Section 3 ●Learning objectives  Test hypotheses regarding two population proportions  Construct and interpret confidence intervals for the difference between two population proportions  Determine the sample size necessary for estimating the difference between two population proportions within a specified margin of error 1 2 3

25. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 25 of 27 Chapter 11 – Section 3 ●We can estimate the required sample sizes to achieve a certain margin of error ●Assuming that the two sample sizes are the same, the margin of error E is

26. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 26 of 27 Chapter 11 – Section 3 ●If p 1 and p 2 are unknown, then the following sample size will always be sufficient ●This is the sample size required for p 1 = p 2 = 0.5

27. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 3 – Slide 27 of 27 Summary: Chapter 11 – Section 3 ●We can compare proportions from two independent samples ●We use a formula with the combined sample sizes and proportions for the standard error ●The overall process, other than the formula for the standard error, are the general hypothesis test and confidence intervals process