CHAPTER 10 Comparing Two Populations or Groups 10.1 Comparing Two Proportions The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers
Comparing Two Proportions Learning Objectives After this section, you should be able to: DESCRIBE the shape, center, and spread of the sampling distribution of the difference of two sample proportions. DETERMINE whether the conditions are met for doing inference about p 1 p 2. CONSTRUCT and INTERPRET a confidence interval to compare two proportions. PERFORM a significance test to compare two proportions. The Practice of Statistics, 5 th Edition 2
Introduction Suppose we want to compare the proportions of individuals with a certain characteristic in Population 1 and Population 2. Let s call these parameters of interest p 1 and p 2. The ideal strategy is to take a separate random sample from each population and to compare the sample proportions with that characteristic. What if we want to compare the effectiveness of Treatment 1 and Treatment 2 in a completely randomized experiment? This time, the parameters p 1 and p 2 that we want to compare are the true proportions of successful outcomes for each treatment. We use the proportions of successes in the two treatment groups to make the comparison. The Practice of Statistics, 5 th Edition 3
The Sampling Distribution of a Difference Between Two Proportions To explore the sampling distribution of the difference between two proportions, let s start with two populations having a known proportion of successes. At School 1, 70% of students did their homework last night At School 2, 50% of students did their homework last night. Suppose the counselor at School 1 takes an SRS of 100 students and records the sample proportion that did their homework. School 2 s counselor takes an SRS of 200 students and records the sample proportion that did their homework. What can we say about the difference ˆ p 1 - ˆ p 2 in the sample proportions? The Practice of Statistics, 5 th Edition 4
The Sampling Distribution of a Difference Between Two Proportions Using Fathom software, we generated an SRS of 100 students from School 1 and a separate SRS of 200 students from School 2. The difference in sample proportions was then be calculated and plotted. We repeated this process 1000 times. What do you notice about the shape, center, and spread of the sampling distribution of p ˆ 1 - p ˆ 2? The Practice of Statistics, 5 th Edition 5
The Sampling Distribution of a Difference Between Two Proportions Both p ˆ 1 and p ˆ 2 are random variables. The statistic p ˆ 1 - p ˆ 2 is the difference of these two random variables. In Chapter 6, we learned that for any two independent random variables X and Y, 2 m X -Y = m X - m Y and s X -Y = s 2 2 X + s Y The Sampling Distribution of the Difference Between Sample Proportions Choose an SRS of size n 1 from Population 1 with proportion of successes p 1 and an independent SRS of size n 2 from Population 2 with proportion of successes p 2. Shape When n 1 p 1, n 1 (1- p 1 ), n 2 p 2 and n 2 (1- p 2 ) are all at least 10, the sampling distribution of p ˆ 1 - p ˆ 2 is approximately Normal. Spread The standard deviation of the sampling distribution of p ˆ 1 - p ˆ 2 is p 1 (1- p 1 ) + p (1- p ) 2 2 n 1 n 2 as long as each sample is no more than 10% of its population (10% condition). The Practice of Statistics, 5 th Edition 6
The Sampling Distribution of a Difference Between Two Proportions The Practice of Statistics, 5 th Edition 7
The Sampling Distribution of a Difference Between Two Proportions Suppose that there are two large high schools, each with more than 2000 students, in a certain town. At School 1, 70% of students did their homework last night. Only 50% of the students at School 2 did their homework last night. The counselor at School 1 takes an SRS of 100 students and records the proportion that did homework. School 2 s counselor takes an SRS of 200 students and records the proportion that did homework a) Describe the shape, center, and spread of the sampling distribution of ˆ p 1 - ˆ p 2. Because n 1 p 1 =100(0.7) = 70, n 1 (1- p 1 ) =100(0.30) = 30, n 2 p 2 = 200(0.5) =100 and n 2 (1- p 2 ) = 200(0.5) =100 are all at least 10, the sampling distribution of p ˆ 1 - p ˆ 2 is approximately Normal. Its mean is p 1 - p 2 = 0.70-0.50 = 0.20. Its standard deviation is 0.7(0.3) 100 + 0.5(0.5) 200 = 0.058. The Practice of Statistics, 5 th Edition 8
Confidence Intervals for p 1 p 2 When data come from two random samples or two groups in a randomized experiment, the statistic p ˆ 1 - p ˆ 2 is our best guess for the value of p 1 -p 2. We can use our familiar formula to calculate a confidence interval for p 1 -p 2 : statistic± (critical value) (standard deviation of statistic) If the Normal condition is met, we find the critical value z* for the given confidence level from the standard Normal curve. The Practice of Statistics, 5 th Edition 9
Confidence Intervals for p 1 p 2 Conditions For Constructing A Confidence Interval About A Difference In Proportions Random: The data come from two independent random samples or from two groups in a randomized experiment. o 10%: When sampling without replacement, check that n 1 (1/10)N 1 and n 2 (1/10)N 2. Because we don't know the values of the parameters p 1 and p 2, we replace them in the standard deviation formula with the sample proportions. The result is the p ˆ standard error of the statistic p ˆ 1 - p ˆ 2 : 1 (1- p ˆ 1 ) p + ˆ 2 (1- p ˆ 2 ) n 1 n 2 The Practice of Statistics, 5 th Edition 10
Confidence Intervals for p 1 p 2 Two-Sample z Interval for a Difference Between Two Proportions The Practice of Statistics, 5 th Edition 11
Significance Tests for p 1 p 2 An observed difference between two sample proportions can reflect an actual difference in the parameters, or it may just be due to chance variation in random sampling or random assignment. Significance tests help us decide which explanation makes more sense. The null hypothesis has the general form H 0 : p 1 - p 2 = hypothesized value We ll restrict ourselves to situations in which the hypothesized difference is 0. Then the null hypothesis says that there is no difference between the two parameters: H 0 : p 1 - p 2 = 0 or, alternatively, H 0 : p 1 = p 2 The alternative hypothesis says what kind of difference we expect. H a : p 1 - p 2 > 0, H a : p 1 - p 2 < 0, or H a : p 1 - p 2 0 The Practice of Statistics, 5 th Edition 12
Significance Tests for p 1 p 2 Conditions For Performing a Significance Test About A Difference In Proportions Random: The data come from two independent random samples or from two groups in a randomized experiment. o 10%: When sampling without replacement, check that n 1 (1/10)N 1 and n 2 (1/10)N 2. The Practice of Statistics, 5 th Edition 13
Significance Tests for p 1 p 2 To do a test, standardize ˆ p 1 - ˆ p 2 to get a z statistic : test statistic = z = statistic - parameter standard deviation of statistic ( p ˆ 1 - p ˆ 2 ) - 0 standard deviation of statistic If H 0 : p 1 = p 2 is true, the two parameters are the same. We call their common value p. We need a way to estimate p, so it makes sense to combine the data from the two samples. This pooled (or combined) sample proportion is: ˆ p C = count of successes in both samples combined count of individuals in both samples combined = X 1 + X 2 n 1 + n 2 The Practice of Statistics, 5 th Edition 14
Significance Tests for p 1 p 2 Two-Sample z Test for the Difference Between Two Proportions The Practice of Statistics, 5 th Edition 15
Inference for Experiments Many important statistical results come from randomized comparative experiments. Defining the parameters in experimental settings is more challenging. Most experiments on people use recruited volunteers as subjects. When subjects are not randomly selected, researchers cannot generalize the results of an experiment to some larger populations of interest. Researchers can draw cause-and-effect conclusions that apply to people like those who took part in the experiment. Unless the experimental units are randomly selected, we don t need to check the 10% condition when performing inference about an experiment. The Practice of Statistics, 5 th Edition 16
Comparing Two Proportions Section Summary In this section, we learned how to DESCRIBE the shape, center, and spread of the sampling distribution of the difference of two sample proportions. DETERMINE whether the conditions are met for doing inference about p 1 p 2. CONSTRUCT and INTERPRET a confidence interval to compare two proportions. PERFORM a significance test to compare two proportions. The Practice of Statistics, 5 th Edition 17