Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section Inference about Two Means: Independent Samples 11.3

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objectives 1.Test hypotheses regarding the difference of two independent means 2.Construct and interpret confidence intervals regarding the difference of two independent means 11-2

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Suppose that a simple random sample of size n 1 is taken from a population with unknown mean μ 1 and unknown standard deviation σ 1. In addition, a simple random sample of size n 2 is taken from a population with unknown mean μ 2 and unknown standard deviation σ 2. If the two populations are normally distributed or the sample sizes are sufficiently large (n 1 ≥ 30, n 2 ≥ 30), then Sampling Distribution of the Difference of Two Means: Independent Samples with Population Standard Deviations Unknown (Welch’s t) 11-3

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. approximately follows Student’s t-distribution with the smaller of n 1 -1 or n 2 -1 degrees of freedom where is the sample mean and s i is the sample standard deviation from population i. Sampling Distribution of the Difference of Two Means: Independent Samples with Population Standard Deviations Unknown (Welch’s t) 11-4

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 1 Test Hypotheses Regarding the Difference of Two Independent Means 11-5

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. To test hypotheses regarding two population means, μ 1 and μ 2, with unknown population standard deviations, we can use the following steps, provided that:  the samples are obtained using simple random sampling;  the samples are independent;  the populations from which the samples are drawn are normally distributed or the sample sizes are large (n 1 ≥ 30, n 2 ≥ 30);  For each sample, the sample size is no more than 5% of the population size. Testing Hypotheses Regarding the Difference of Two Means 11-6

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 1: Determine the null and alternative hypotheses. The hypotheses are structured in one of three ways: 11-7 Step 2: Select a level of significance, α, based on the seriousness of making a Type I error. Step 3: Compute the test statistic Use Table VI to determine the critical value using the smaller of n 1 – 1 or n 2 – 1 degrees of freedom. Step 4: Find the critical value(s).

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Compare the critical value with the test statistic: Classical Approach 11-8

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Use Table VI to determine the P-value using the smaller of n 1 – 1 or n 2 – 1 degrees of freedom. P-Value Approach 11-9 Technology: Use a statistical spreadsheet or calculator with statistical capabilities to obtain the P-value. The directions for obtaining the P-value using the TI-83/84 Plus graphing calculator, Excel, MINITAB, and StatCrunch are in the Technology Step-by-Step in the text. Finally: State the conclusion.

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. These procedures are robust, which means that minor departures from normality will not adversely affect the results. However, if the data have outliers, the procedure should not be used

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. A researcher wanted to know whether “state” quarters had a weight that is more than “traditional” quarters. He randomly selected 18 “state” quarters and 16 “traditional” quarters, weighed each of them and obtained the following data. Parallel Example 1: Testing Hypotheses Regarding Two Means 11-11

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Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Test the claim that “state” quarters have a mean weight that is more than “traditional” quarters at the α = 0.05 level of significance. NOTE: A normal probability plot of “state” quarters indicates the population could be normal. A normal probability plot of “traditional” quarters indicates the population could be normal 11-13

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. No outliers

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 1: We want to determine whether state quarters weigh more than traditional quarters: H 0 : μ 1 = μ 2 versus H 1 : μ 1 > μ 2 Step 2: The level of significance is α = Step 3: The test statistic is Solution 11-15

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Solution: Classical Approach This is a right-tailed test with α = Since n 1 – 1 = 17 and n 2 – 1 = 15, we will use 15 degrees of freedom. The corresponding critical value is t 0.05 =

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 4: Since the test statistic, t 0 = 2.53 is greater than the critical value t.05 = 1.753, we reject the null hypothesis. Solution: Classical Approach 11-17

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Because this is a right-tailed test, the P-value is the area under the t-distribution to the right of the test statistic t 0 = That is, P-value = P(t > 2.53) ≈ Solution: P-Value Approach 11-18

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 4: Since the P-value is less than the level of significance α = 0.05, we reject the null hypothesis. Solution: P-Value Approach 11-19

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 5: There is sufficient evidence at the α = 0.05 level to conclude that the state quarters weigh more than the traditional quarters. Solution 11-20

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. NOTE: The degrees of freedom used to determine the critical value in the last example are conservative. Results that are more accurate can be obtained by using the following degrees of freedom: 11-21

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 2 Construct and Interpret Confidence Intervals Regarding the Difference of Two Independent Means 11-22

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. A simple random sample of size n 1 is taken from a population with unknown mean μ 1 and unknown standard deviation σ 1. Also, a simple random sample of size n 2 is taken from a population with unknown mean μ 2 and unknown standard deviation σ 2. If the two populations are normally distributed or the sample sizes are sufficiently large (n 1 ≥ 30 and n 2 ≥ 30), a (1 – α)100% confidence interval about μ 1 – μ 2 is given by... Constructing a (1 – α)100% Confidence Interval for the Difference of Two Means 11-23

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Lower bound: and Upper bound: where t α/2 is computed using the smaller of n 1 – 1 or n 2 – 1 degrees of freedom or Formula (2). Constructing a (1 – α)100% Confidence Interval for the Difference of Two Means 11-24

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Construct a 95% confidence interval about the difference between the population mean weight of a “state” quarter versus the population mean weight of a “traditional” quarter. Parallel Example 3: Constructing a Confidence Interval for the Difference of Two Means 11-25

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. We have already verified that the populations are approximately normal and that there are no outliers. Recall = 5.702, s 1 = , = and s 2 = From Table VI with α = 0.05 and 15 degrees of freedom, we find t α/2 = Solution 11-26

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Thus, Lower bound = Upper bound = Solution 11-27

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. We are 95% confident that the mean weight of the “state” quarters is between and ounces more than the mean weight of the “traditional” quarters. Since the confidence interval does not contain 0, we conclude that the “state” quarters weigh more than the “traditional” quarters. Solution 11-28

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. When the population variances are assumed to be equal, the pooled t-statistic can be used to test for a difference in means for two independent samples. The pooled t- statistic is computed by finding a weighted average of the sample variances and using this average in the computation of the test statistic. The advantage to this test statistic is that it exactly follows Student’s t-distribution with n 1 +n 2 -2 degrees of freedom. The disadvantage to this test statistic is that it requires that the population variances be equal

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section Inference about Two Population Standard Deviations 11.4

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objectives 1.Find critical values of the F-distribution 2.Test hypotheses regarding two population standard deviations 11-31

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 1 Find Critical Values of the F-distribution 11-32

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Requirements for Testing Claims Regarding Two Population Standard Deviations 1. The samples are independent simple random samples

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Requirements for Testing Claims Regarding Two Population Standard Deviations 1. The samples are independent simple random samples. 2. The populations from which the samples are drawn are normally distributed

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. CAUTION! If the populations from which the samples are drawn are not normal, do not use the inferential procedures discussed in this section

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Notation Used When Comparing Two Population Standard Deviations : Variance for population 1 : Variance for population 2 : Sample variance for population 1 : Sample variance for population 2 n 1 : Sample size for population 1 n 2 : Sample size for population

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Fisher's F-distribution If and and are sample variances from independent simple random samples of size n 1 and n 2, respectively, drawn from normal populations, then follows the F-distribution with n 1 – 1 degrees of freedom in the numerator and n 2 – 1 degrees of freedom in the denominator

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Characteristics of the F-distribution 1. It is not symmetric. The F-distribution is skewed right

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Characteristics of the F-distribution 1. It is not symmetric. The F-distribution is skewed right. 2. The shape of the F-distribution depends upon the degrees of freedom in the numerator and denominator. This is similar to the χ 2 distribution and Student’s t- distribution, whose shapes depend upon their degrees of freedom

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Characteristics of the F-distribution 1. It is not symmetric. The F-distribution is skewed right. 2. The shape of the F-distribution depends upon the degrees of freedom in the numerator and denominator. This is similar to the χ 2 distribution and Student’s t- distribution, whose shape depends upon their degrees of freedom. 3. The total area under the curve is

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Characteristics of the F-distribution 1. It is not symmetric. The F-distribution is skewed right. 2. The shape of the F-distribution depends upon the degrees of freedom in the numerator and denominator. This is similar to the χ 2 distribution and Student’s t- distribution, whose shape depends upon their degrees of freedom. 3. The total area under the curve is The values of F are always greater than or equal to zero

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Copyright © 2013, 2010 and 2007 Pearson Education, Inc. is the critical F with n 1 – 1 degrees of freedom in the numerator and n 2 – 1 degrees of freedom in the denominator and an area of α to the right of the critical F

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. To find the critical F with an area of α to the left, use the following: 11-44

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Find the critical F-value: a)for a right-tailed test with α = 0.1, degrees of freedom in the numerator = 8 and degrees of freedom in the denominator = 4. b)for a two-tailed test with α = 0.05, degrees of freedom in the numerator = 20 and degrees of freedom in the denominator = 15. Parallel Example 1: Finding Critical Values for the F-distribution 11-45

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. a)F 0.1,8,4 = 3.95 b)F.025,20,15 = 2.76 Solution 11-46

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. NOTE: If the number of degrees of freedom is not found in the table, we follow the practice of choosing the degrees of freedom closest to that desired. If the degrees of freedom is exactly between two values, find the mean of the values

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 2 Test Hypotheses Regarding Two Population Standard Deviations 11-48

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. To test hypotheses regarding two population standard deviations, σ 1 and σ 2, we can use the following steps, provided that 1.the samples are obtained using simple random sampling, 2.the sample data are independent, and 3.the populations from which the samples are drawn are normally distributed. Test Hypotheses Regarding Two Population Standard Deviations 11-49

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 1: Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways: Two-TailedLeft-TailedRight-Tailed H 0 : σ 1 = σ 2 H 1 : σ 1 ≠ σ 2 H 1 : σ 1 < σ 2 H 1 : σ 1 > σ 2 Note: σ 1 is the population standard deviation for population 1 and σ 2 is the population standard deviation for population

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 2: Select a level of significance, α, based on the seriousness of making a Type I error

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 3: Compute the test statistic which follows Fisher’s F-distribution with n 1 – 1 degrees of freedom in the numerator and n 2 – 1 degrees of freedom in the denominator

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Use Table VIII to determine the critical value(s) using n 1 – 1 degrees of freedom in the numerator and n 2 – 1 degrees of freedom in the denominator. Classical Approach 11-53

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Classical Approach (critical value) Two-Tailed 11-54

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Classical Approach (critical value) Left-Tailed 11-55

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Classical Approach (critical value) Right-Tailed 11-56

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 4: Compare the critical value with the test statistic: Classical Approach Two-TailedLeft-TailedRight-Tailed If or, reject the null hypothesis. If, reject the null hypothesis. If, reject the null hypothesis

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 3: Use technology to determine the P-value. P-Value Approach Step 4: If P-value < α, reject the null hypothesis. Step 5: State the conclusion.

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. CAUTION! The procedures just presented are not robust, minor departures from normality will adversely affect the results of the test. Therefore, the test should be used only when the requirement of normality has been verified

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. A researcher wanted to know whether “state” quarters had a standard deviation weight that is less than “traditional” quarters. He randomly selected 18 “state” quarters and 16 “traditional” quarters, weighed each of them and obtained the data on the next slide. A normal probability plot indicates that the sample data could come from a population that is normal. Test the researcher’s claim at the α = 0.05 level of significance. Parallel Example 2: Testing Hypotheses Regarding Two Population Standard Deviations 11-60

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Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 1: The researcher wants to know if “state” quarters have a standard deviation weight that is less than “traditional” quarters. Thus H 0 : σ 1 = σ 2 versus H 1 : σ 1 < σ 2 This is a left-tailed test. Step 2: The level of significance is α = Solution 11-62

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 3: The standard deviation of “state” quarters was found to be and the standard deviation of “traditional” quarters was found to be The test statistic is then Solution 11-63

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Since this is a left-tailed test, we determine the critical value at the 1 – α = 1 – 0.05 = 0.95 level of significance with n 1 – 1=18 – 1=17 degrees of freedom in the numerator and n 2 – 1 = 16 – 1 = 15 degrees of freedom in the denominator. Thus, Note: we used the table value F 0.05,15,15 for the above calculation since this is the closest to the required degrees of freedom available from Table VIII. Solution: Classical Approach 11-64

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 4: Since the test statistic F 0 = 0.52 is greater than the critical value F 0.95,17,15 =0.42, we fail to reject the null hypothesis. Solution: Classical Approach 11-65

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 3: Using technology, we find that the P-value is If the statement in the null hypothesis were true, we would expect to get the results obtained about 10 out of 100 times. This is not very unusual. Solution: P-Value Approach 11-66

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 4: Since the P-value is greater than the level of significance, α = 0.05, we fail to reject the null hypothesis. Solution: P-Value Approach 11-67

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Step 5: There is not enough evidence to conclude that the standard deviation of weight is less for “state” quarters than it is for “traditional” quarters at the α = 0.05 level of significance. Solution 11-68

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section Putting It Together: Which Method Do I Use? 11.5

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 1.Determine the appropriate hypothesis test to perform 11-70

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 1 Determine the Appropriate Hypothesis Test to Perform 11-71

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. What parameter is addressed in the hypothesis? Proportion, p σ or σ 2 Mean, μ 11-72

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Proportion, p Is the sampling Dependent or Independent? 11-73

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Proportion, p Dependent samples: Provided the samples are obtained randomly and the total number of observations where the outcomes differ is at least 10, use the normal distribution with 11-74

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Proportion, p Independent samples: Provided for each sample and the sample size is no more than 5% of the population size, use the normal distribution with where 11-75

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. σ or σ 2 Provided the data are normally distributed, use the F-distribution with 11-76

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Mean, μ Is the sampling Dependent or Independent? 11-77

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Mean, μ Dependent samples: Provided each sample size is greater than 30 or the differences come from a population that is normally distributed, use Student’s t-distribution with n-1 degrees of freedom with 11-78

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Mean, μ Independent samples: Provided each sample size is greater than 30 or each population is normally distributed, use Student’s t-distribution 11-79

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