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

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

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

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 …

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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 and 48, so our method can be used

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

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

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

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

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

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

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

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