Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series by Mario F. Triola

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Chapter 9 Inferences from Two Samples 9-1 Review and Preview 9-2 Two Proportions 9-3 Two Means: Independent Samples 9-4 Two Dependent Samples (Matched Pairs) 9-5 Two Variances or Standard Deviations

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Key Concept In this section we present methods for (1) testing a claim made about two population proportions and (2) constructing a confidence interval estimate of the difference between the two population proportions. This section is based on proportions, but we can use the same methods for dealing with probabilities or the decimal equivalents of percentages.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. (the sample proportion) The corresponding notations apply to which come from population 2. Notation for Two Proportions For population 1, we let: = population proportion = size of the sample = number of successes in the sample

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc.  The pooled sample proportion is given by: Pooled Sample Proportion

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Requirements 1. We have proportions from two independent simple random samples. 2. For each of the two samples, the number of successes is at least 5 and the number of failures is at least 5.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Test Statistic for Two Proportions For where (assumed in the null hypothesis)

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Testing Two Proportions P-value: P-values are automatically provided by technology. If technology is not available, use Table A-2. Critical values: Use Table A-2. (Based on the significance level α, find critical values by using the procedures introduced in Section 8-2 in the text.)

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Confidence Interval Estimate of p 1 – p 2

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example Do people having different spending habits depending on the type of money they have? 89 undergraduates were randomly assigned to two groups and were given a choice of keeping the money or buying gum or mints. The claim is that “money in large denominations is less likely to be spent relative to an equivalent amount in many smaller denominations”. Let’s test the claim at the 0.05 significance level.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example Below are the sample data and summary statistics: Group 1Group 2 Subjects Given $1 Bill Subjects Given 4 Quarters Spent the moneyx 1 = 12x 2 = 27 Subjects in groupn 1 = 46n 2 = 43

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example Requirement Check: 1.The 89 subjects were randomly assigned to two groups, so we consider these independent random samples. 2.The subjects given the $1 bill include 12 who spent it and 34 who did not. The subjects given the quarters include 27 who spent it and 16 who did not. All counts are above 5, so the requirements are all met.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example Step 1: The claim that “money in large denominations is less likely to be spent” can be expressed as p 1 < p 2. Step 2: If p 1 < p 2 is false, then p 1 ≥ p 2. Step 3: The hypotheses can be written as:

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example Step 4: The significance level is α = Step 5: We will use the normal distribution to run the test with:

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example Step 6: Calculate the value of the test statistic:

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example Step 6: This is a left-tailed test, so the P-value is the area to the left of the test statistic z = –3.49, or The critical value is also shown below.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example Step 7: Because the P-value of is less than the significance level of α = 0.05, reject the null hypothesis. There is sufficient evidence to support the claim that people with money in large denominations are less likely to spend relative to people with money in smaller denominations. It should be noted that the subjects were all undergraduates and care should be taken before generalizing the results to the general population.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example We can also construct a confidence interval to estimate the difference between the population proportions. Caution: The confidence interval uses standard deviations based on estimated values of the population proportions, and consequently, a conclusion based on a confidence interval might be different from a conclusion based on a hypothesis test.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example Construct a 90% confidence interval estimate of the difference between the two population proportions. What does the result suggest about our claim about people spending large denominations relative to spending small denominations?

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example The confidence interval limits do not include 0, implying that there is a significant difference between the two proportions. There does appear to be sufficient evidence to support the claim that “money in large denominations is less likely to be spent relative to an equivalent amount in many smaller denominations.”

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Why Do the Procedures of This Section Work? The distribution of can be approximated by a normal distribution with mean p 1, and standard deviation: The difference can be approximated by a normal distribution with mean p 1 – p 2 and variance The variance of the differences between two independent random variables is the sum of their individual variances.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Why Do the Procedures of This Section Work? The preceding variance leads to We now know that the distribution of is approximately normal, with mean and standard deviation as shown above, so the z test statistic has the form given earlier.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Why Do the Procedures of This Section Work? When constructing the confidence interval estimate of the difference between two proportions, we don’t assume that the two proportions are equal, and we estimate the standard deviation as

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Why Do the Procedures of This Section Work? In the test statistic use the positive and negative values of z (for two tails) and solve for The results are the limits of the confidence interval given earlier.