Hypothesis Testing with Two Samples, Study notes of Statistics

Download Hypothesis Testing with Two Samples and more Study notes Statistics in PDF only on Docsity! 1 Hypothesis Testing with Two Samples Chapter 8 § 8.1 Testing the Difference Between Means (Large Independent Samples) Larson & Farber, Elementary Statistics: Picturing the World, 3e 3 Two Sample Hypothesis Testing In a two-sample hypothesis test, two parameters from two populations are compared. For a two-sample hypothesis test, 1. the null hypothesis H0 is a statistical hypothesis that usually states there is no difference between the parameters of two populations. The null hypothesis always contains the symbol ≤, =, or ≥. 2. the alternative hypothesis Ha is a statistical hypothesis that is true when H0 is false. The alternative hypothesis always contains the symbol >, ≠, or <. 2 Larson & Farber, Elementary Statistics: Picturing the World, 3e 4 Two Sample Hypothesis Testing To write a null and alternative hypothesis for a two-sample hypothesis test, translate the claim made about the population parameters from a verbal statement to a mathematical statement. H0: µ1 = µ2 Ha: µ1 ≠ µ2 H0: µ1 ≤ µ2 Ha: µ1 > µ2 H0: µ1 ≥ µ2 Ha: µ1 < µ2 Regardless of which hypotheses used, µ1 = µ2 is always assumed to be true. Larson & Farber, Elementary Statistics: Picturing the World, 3e 5 Two Sample z-Test Three conditions are necessary to perform a z-test for the difference between two population means µ1 and µ2. 1. The samples must be randomly selected. 2. The samples must be independent. Two samples are independent if the sample selected from one population is not related to the sample selected from the second population. 3. Each sample size must be at least 30, or, if not, each population must have a normal distribution with a known standard deviation. Larson & Farber, Elementary Statistics: Picturing the World, 3e 6 Two Sample z-Test If these requirements are met, the sampling distribution for (the difference of the sample means) is a normal distribution with mean and standard error of 1 2 1 2 1 2x x x xµ µ µ µ µ− = − = − 1 2x x− and 1 2 1 2 2 2 2 2 1 2 1 2 .x x x x σ σ σ σ σ n n− = + = + 1 2µ µ− 1 2 x xσ −− 1 2 x xσ − Sampling distribution for 1 2x x− 1 2x x− 5 Larson & Farber, Elementary Statistics: Picturing the World, 3e 13 Two Sample t-Test 1. The samples must be randomly selected. 2. The samples must be independent. Two samples are independent if the sample selected from one population is not related to the sample selected from the second population. 3. Each population must have a normal distribution. If samples of size less than 30 are taken from normally-distributed populations, a t-test may be used to test the difference between the population means µ1 and µ2. Three conditions are necessary to use a t-test for small independent samples. Larson & Farber, Elementary Statistics: Picturing the World, 3e 14 Two Sample t-Test Two-Sample t-Test for the Difference Between Means A two-sample t-test is used to test the difference between two population means µ1 and µ2 when a sample is randomly selected from each population. Performing this test requires each population to be normally distributed, and the samples should be independent. The standardized test statistic is If the population variances are equal, then information from the two samples is combined to calculate a pooled estimate of the standard deviation ( ) ( ) 1 2 1 2 1 2 . x x x x µ µ t σ − − − − = σ̂. ( ) ( )2 2 1 1 2 2 1 2 1 1 ˆ 2 n s n s σ n n − + − = + − Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3e 15 Two Sample t-Test Two-Sample t-Test (Continued) The standard error for the sampling distribution of is and d.f.= n1 + n2 – 2. If the population variances are not equal, then the standard error is and d.f = smaller of n1 – 1 or n2 – 1. 1 2 1 2 1 1 ˆx xσ σ n n− = ⋅ + 1 2x x− Variances equal 1 2 2 2 1 2 1 2 x x s s σ n n− = + Variances not equal 6 Larson & Farber, Elementary Statistics: Picturing the World, 3e 16 Normal or t-Distribution? Are both sample sizes at least 30? Are both populations normally distributed? You cannot use the z- test or the t-test. No Yes Are both population standard deviations known? Use the z-test.Yes No Are the population variances equal? Use the z-test. Use the t-test with and d.f = smaller of n1 – 1 or n2 – 1. 1 2 2 2 1 2 1 2 x x s s σ n n− = + Use the t-test with and d.f = n1 + n2 – 2. 1 2 1 2 1 1 ˆx xσ σ n n− = + Yes No No Yes Larson & Farber, Elementary Statistics: Picturing the World, 3e 17 Two Sample t-Test for the Means 1. State the claim mathematically. Identify the null and alternative hypotheses. 2. Specify the level of significance. 3. Identify the degrees of freedom and sketch the sampling distribution. 4. Determine the critical value(s). Continued. Using a Two-Sample t-Test for the Difference Between Means (Small Independent Samples) In Words In Symbols State H0 and Ha. Identify α. Use Table 5 in Appendix B. d.f. = n1+ n2 – 2 or d.f. = smaller of n1 – 1 or n2 – 1. Larson & Farber, Elementary Statistics: Picturing the World, 3e 18 Two Sample t-Test for the Means In Words In Symbols If t is in the rejection region, reject H0. Otherwise, fail to reject H0. 5. Determine the rejection regions(s). 6. Find the standardized test statistic. 7. Make a decision to reject or fail to reject the null hypothesis. 8. Interpret the decision in the context of the original claim. Using a Two-Sample t-Test for the Difference Between Means (Small Independent Samples) ( ) ( ) 1 2 1 2 1 2 x x x x µ µ t σ − − − − = 7 Larson & Farber, Elementary Statistics: Picturing the World, 3e 19 Two Sample t-Test for the Means Example: A random sample of 17 police officers in Brownsville has a mean annual income of $35,800 and a standard deviation of $7,800. In Greensville, a random sample of 18 police officers has a mean annual income of $35,100 and a standard deviation of $7,375. Test the claim at α = 0.01 that the mean annual incomes in the two cities are not the same. Assume the population variances are equal. Ha: µ1 ≠ µ2 (Claim) H0: µ1 = µ2 Continued. –t0 = –2.576 d.f. = n1 + n2 – 2 = 17 + 18 – 2 = 33 t0 = 2.576 t 0 1 2 3-3 -2 -1 005.0 2 1 =α 005.0 2 1 =α Larson & Farber, Elementary Statistics: Picturing the World, 3e 20 Two Sample t-Test for the Means Example continued: The standardized error is 1 2 1 2 1 1 ˆx xσ σ n n− = + ( ) ( )2 217 1 7800 18 1 7375 1 1 17 18 2 17 18 − + − = ⋅ + + − ( ) ( )2 2 1 1 2 2 1 2 1 2 1 1 1 1 2 n s n s n n n n − + − = ⋅ + + − Ha: µ1 ≠ µ2 (Claim) H0: µ1 = µ2 –t0 = –2.576 t 0 1 2 3-3 -2 -1 t0 = 2.576 7584.0355(0.3382)≈ Continued.2564.92≈ Larson & Farber, Elementary Statistics: Picturing the World, 3e 21 Two Sample t-Test for the Means ( ) ( ) 1 2 1 2 1 2 x x x x µ µ t σ − − − − = Example continued: The standardized test statistic is 0.273≈ Fail to reject H0. There is not enough evidence at the 1% level to support the claim that the mean annual incomes differ. Ha: µ1 ≠ µ2 (Claim) H0: µ1 = µ2 –t0 = –2.576 t 0 1 2 3-3 -2 -1 t0 = 2.576 ( )35800 35100 0 2564.92 − − = 10 Larson & Farber, Elementary Statistics: Picturing the World, 3e 28 t-Test for the Difference Between Means t-Test for the Difference Between Means A t-test can be used to test the difference of two population means when a sample is randomly selected from each population. The requirements for performing the test are that each population must be normal and each member of the first sample must be paired with a member of the second sample. The test statistic is and the standardized test statistic is The degrees of freedom are d.f. = n – 1. .d d d µ t s n − = d d n ∑= Larson & Farber, Elementary Statistics: Picturing the World, 3e 29 t-Test for the Difference Between Means 1. State the claim mathematically. Identify the null and alternative hypotheses. 2. Specify the level of significance. 3. Identify the degrees of freedom and sketch the sampling distribution. 4. Determine the critical value(s). Continued. Using the t-Test for the Difference Between Means (Dependent Samples) In Words In Symbols State H0 and Ha. Identify α. Use Table 5 in Appendix B. d.f. = n – 1 Larson & Farber, Elementary Statistics: Picturing the World, 3e 30 t-Test for the Difference Between Means In Words In Symbols 5. Determine the rejection region(s). 6. Calculate and Use a table. 7. Find the standardized test statistic. Using a Two-Sample t-Test for the Difference Between Means (Small Independent Samples) d d n ∑=d .ds 2 2( ) ( ) ( 1)d n d d s n n ∑ − ∑= − d d d µ t s n − = 11 Larson & Farber, Elementary Statistics: Picturing the World, 3e 31 t-Test for the Difference Between Means In Words In Symbols If t is in the rejection region, reject H0. Otherwise, fail to reject H0. 8. Make a decision to reject or fail to reject the null hypothesis. 9. Interpret the decision in the context of the original claim. Using a Two-Sample t-Test for the Difference Between Means (Small Independent Samples) Larson & Farber, Elementary Statistics: Picturing the World, 3e 32 t-Test for the Difference Between Means Example: A reading center claims that students will perform better on a standardized reading test after going through the reading course offered by their center. The table shows the reading scores of 6 students before and after the course. At α = 0.05, is there enough evidence to conclude that the students’ scores after the course are better than the scores before the course? Continued. 899286898588Score (after) 788176709685Score (before) 654321Student Ha: µd > 0 (Claim) H0: µd ≤ 0 Larson & Farber, Elementary Statistics: Picturing the World, 3e 33 t-Test for the Difference Between Means 899286898588Score (after) −11−11−10−1911−3d 1211211003611219d 2 788176709685Score (before) 654321Student Example continued: Ha: µd > 0 (Claim) H0: µd ≤ 0 Continued. d.f. = 6 – 1 = 5 t0 = 2.015 t 0 1 2 3-3 -2 -1 α = 0.05 d d n ∑= 43 7.167 6 −= ≈ − 6(833) 1849 6(5) −= 104.967≈ 10.245≈ 2 2( ) ( ) ( 1)d n d d s n n ∑ − ∑= − 43d∑ = − 2 833d∑ = d = (score before) – (score after) 12 Larson & Farber, Elementary Statistics: Picturing the World, 3e 34 t-Test for the Difference Between Means Example continued: Ha: µd > 0 (Claim) H0: µd ≤ 0 t0 = 2.015 t 0 1 2 3-3 -2 -1 Fail to reject H0. There is not enough evidence at the 5% level to support the claim that the students’ scores after the course are better than the scores before the course. d d d µ t s n − = The standardized test statistic is 7.167 0 10.245 6 − −= 1.714.≈ − § 8.4 Testing the Difference Between Proportions Larson & Farber, Elementary Statistics: Picturing the World, 3e 36 Two Sample z-Test for Proportions A z-test is used to test the difference between two population proportions, p1 and p2. Three conditions are required to conduct the test. 1. The samples must be randomly selected. 2. The samples must be independent. 3. The samples must be large enough to use a normal sampling distribution. That is, n1p1 ≥ 5, n1q1 ≥ 5, n2p2 ≥ 5, and n2q2 ≥ 5.