Comparing Two Proportions BPS chapter 21 © 2006 W.H. Freeman and Company

Parameter of interest When comparing two proportions from two populations or two treatments, what is the parameter of interest?

Parameter of interest (answer) When comparing two proportions from two populations or two treatments, what is the parameter of interest?

Confidence interval The purpose of a confidence interval comparing two proportions (from two populations or two treatments) is to give a range of reasonable values for the Level of confidence. Overall sample proportion. Overall population proportion. Difference between p1 and p2. Values for the sum of p1 and p2.

Confidence interval (answer) The purpose of a confidence interval comparing two proportions (from two populations or two treatments) is to give a range of reasonable values for the Level of confidence. Overall sample proportion. Overall population proportion. Difference between p1 and p2. Values for the sum of p1 and p2.

Sampling distribution What is the mean of the sampling distribution of ? p1 p2 p1 - p2

Sampling distribution (answer) What is the mean of the sampling distribution of ? p1 p2 p1 - p2

Sampling distribution What is the shape of the sampling distribution of , when all conditions are met? Normal Approximately normal Right-skewed Left-skewed

Sampling distribution (answer) What is the shape of the sampling distribution of , when all conditions are met? Normal Approximately normal Right-skewed Left-skewed

Hypothesis test Suppose we want to test whether the proportions from two different populations are significantly different from each other. What are the appropriate null and alternative hypotheses?

Hypothesis test (answer) Suppose we want to test whether the proportions from two different populations are significantly different from each other. What are the appropriate null and alternative hypotheses?

Pooled sample proportion When do we use the pooled sample proportion? When doing a confidence interval for p1 – p2. When doing a hypothesis test of .

Pooled sample proportion (answer) When do we use the pooled sample proportion? When doing a confidence interval for p1 – p2. When doing a hypothesis test of .

Hypothesis testing You have available data showing that 70% of all eligible students in Pennsylvania and 70% of all eligible students in Rhode Island took the SAT during the 1994-1995 school year. You are interested in testing whether the proportion of eligible students in Pennsylvania (p1) who plan to take the SAT during the 2004-2005 school year is significantly different from the proportion of eligible students in Rhode Island (p2) who plan to take the SAT during the 2004-2005 school year. Which of the following pair of hypotheses is appropriate for this test?

Hypothesis testing (answer) You have available data showing that 70% of all eligible students in Pennsylvania and 70% of all eligible students in Rhode Island took the SAT during the 1994-1995 school year. You are interested in testing whether the proportion of eligible students in Pennsylvania (p1) who plan to take the SAT during the 2004-2005 school year is significantly different from the proportion of eligible students in Rhode Island (p2) who plan to take the SAT during the 2004-2005 school year. Which of the following pair of hypotheses is appropriate for this test?

Sampling distribution Suppose you take an SRS of size 1000 of Pennsylvania students (p1 ) eligible to take the SAT and find that 75% plan to take the SAT during the 2004-2005 school year. You also take an SRS of size 1000 of Rhode Island students (p2 ) eligible to take the SAT and find that 76% plan to take the SAT during the 2004-2005 school year. What is the mean of the sampling distribution of under the null hypothesis ? 0.76 – 0.75 = 0.01 0.75 – 0.76 = -0.01 0.75 0.76 Cannot be determined from the information given.

Sampling distribution (answer) Suppose you take an SRS of size 1000 of Pennsylvania students (p1 ) eligible to take the SAT and find that 75% plan to take the SAT during the 2004-2005 school year. You also take an SRS of size 1000 of Rhode Island students (p2 ) eligible to take the SAT and find that 76% plan to take the SAT during the 2004-2005 school year. What is the mean of the sampling distribution of under the null hypothesis ? 0.76 – 0.75 = 0.01 0.75 – 0.76 = -0.01 0.75 0.76 Cannot be determined from the information given.

Hypothesis testing If you calculate a 95% confidence interval for the difference in the proportion of eligible students in Pennsylvania and Rhode Island that plan to take the SAT during the 2004-2005 school year to be (-0.048, 0.028), what is your conclusion to the two-sided test with null hypothesis ? Is the test statistically significant? Yes, because 0 is included in the interval. Yes, because 0 is not included in the interval. No, because 0 is included in the interval. No, because 0 is not included in the interval.

Hypothesis testing (answer) If you calculate a 95% confidence interval for the difference in the proportion of eligible students in Pennsylvania and Rhode Island that plan to take the SAT during the 2004-2005 school year to be (-0.048, 0.028), what is your conclusion to the two sided-test with null hypothesis ? Is the test statistically significant? Yes, because 0 is included in the interval. Yes, because 0 is not included in the interval. No, because 0 is included in the interval. No, because 0 is not included in the interval.