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. 9-1 Review In Chapters 7 and 8 we introduced methods of inferential statistics. In Chapter 7 we presented methods of constructing confidence interval estimates of population parameters. In Chapter 8 we presented methods of testing claims made about population parameters. Chapters 7 and 8 both involved methods for dealing with a sample from a single population.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. 9-1 Preview The objective of this chapter is to extend the methods for estimating values of population parameters and the methods for testing hypotheses to situations involving two sets of sample data instead of just one. We will present methods for using sample data from two populations so that inferences can be made about those populations.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. 9-2 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. 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 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 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.” 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 Review We began a study of inferential statistics in Chapter 7 when we presented methods for estimating a parameter for a single population and in Chapter 8 when we presented methods of testing claims about a single population. In Chapter 9 we extended those methods to situations involving two populations. In Chapter 10 we considered methods of correlation and regression using paired sample data.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc Preview  We focus on analysis of categorical (qualitative or attribute) data that can be separated into different categories.  Hypothesis test: Observed counts agree with some claimed distribution.  The contingency table or two-way frequency table (two or more rows and columns).  Use the χ 2 (chi-square) distribution.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Key Concept In this section, we consider sample data consisting of observed frequency counts arranged in a single row or column (called a one-way frequency table). We will use a hypothesis test for the claim that the observed frequency counts agree with some claimed distribution, so that there is a good fit of the observed data with the claimed distribution.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Definition A goodness-of-fit test is used to test the hypothesis that an observed frequency distribution fits (or conforms to) some claimed distribution.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Notation O represents the observed frequency of an outcome, found from the sample data. E represents the expected frequency of an outcome, found by assuming that the distribution is as claimed. k represents the number of different categories or cells. n represents the total number of trials. Goodness-of-Fit Test

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Goodness-of-Fit Test 1.The data have been randomly selected. 2.The sample data consist of frequency counts for each of the different categories. 3.For each category, the expected frequency is at least 5. (The expected frequency for a category is the frequency that would occur if the data actually have the distribution that is being claimed. There is no requirement that the observed frequency for each category must be at least 5.) Requirements

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Goodness-of-Fit Hypotheses and Test Statistic

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. P-Values and Critical Values P-Values P-values are typically provided by technology, or a range of P-values can be found from Table A-4. Critical Values 1. Found in Table A-4 using k – 1 degrees of freedom, where k = number of categories. 2. Goodness-of-fit hypothesis tests are always right-tailed.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. A close agreement between observed and expected values will lead to a small value of χ 2 and a large P-value. A large disagreement between observed and expected values will lead to a large value of χ 2 and a small P-value. A significantly large value of χ 2 will cause a rejection of the null hypothesis of no difference between the observed and the expected. Goodness-of-Fit Test

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Goodness- Of-Fit Tests

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc... Example A random sample of 100 weights of Californians is obtained, and the last digit of those weights are summarized on the next slide. When obtaining weights, it is extremely important to actually measure the weights rather than ask people to self-report them. By analyzing the last digit, we can verify the weights were actually measured since reported weights tend to be rounded to something ending with a 0 or a 5. Test the claim that the sample is from a population of weights in which the last digits do not occur with the same frequency.

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

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc... Example - Continued Requirement Check: 1.The data come from randomly selected subjects. 2.The data do consist of counts. 3.With 100 sample values and 10 categories that are claimed to be equally likely, each expected frequency is 10, which is greater than 5. All requirements are met to proceed.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc... Example - Continued Step 1: The original claim is that the digits do not occur with the same frequency. That is: Step 2: If the original claim is false, then all the probabilities are the same:

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc... Example - Continued Step 3: The hypotheses can be written as: Step 4: No significance level was specified, so we select α = Step 5: We use the goodness-of-fit test with a χ 2 distribution.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc... Example - Continued Step 6: The calculation of the test statistic is given:

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc... Example - Continued Step 6: The test statistic is χ 2 = and the critical value is χ 2 = (Table A-4). The P-value was found to be less than using technology.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc... Example - Continued Step 7: Reject the null hypothesis, since the P-value is small and the test statistic is in the critical region. Step 8: We conclude there is sufficient evidence to support the claim that the last digits do not occur with the same relative frequency. In other words, we have evidence that the weights were self-reported by the subjects, and the subjects were not actually weighed.