1. Objective: To test claims about inferences for two sample means, under specific conditions

2. COMPARING TWO MEANS

3. Because we are working with means and estimating the standard error of their difference using the data, we shouldn’t be surprised that the sampling model is a Student’s t. two-sample t-interval The confidence interval we build is called a two-sample t-interval (for the difference in means). two-sample t-test The corresponding hypothesis test is called a two-sample t-test. COMPARING TWO MEANS (CONT.)

4. Independence Assumption Independence Assumption (Each condition needs to be checked for both groups.): Randomization Condition Randomization Condition: Were the data collected with suitable randomization (representative random samples or a randomized experiment)? 10% Condition 10% Condition: We don’t usually check this condition for differences of means. We will check it for means only if we have a very small population or an extremely large sample. TWO-SAMPLE T-INTERVAL AND T- TEST CONDITIONS

5. Normal Population Assumption: Normal Population Assumption: Nearly Normal Condition: Nearly Normal Condition: This must be checked for both groups. A violation by either one violates the condition. Independent Groups Assumption: Independent Groups Assumption: The two groups we are comparing must be independent of each other. (See 8.3 if the groups are not independent of one another…) TWO-SAMPLE T-INTERVAL AND T- TEST CONDITIONS (CONT.)

6. TWO-SAMPLE T-INTERVAL

7. The special formula for the degrees of freedom for our t critical value is a bear: Because of this, we will let technology calculate degrees of freedom for us! DEGREES OF FREEDOM

8. 1.Check Conditions and show that you have checked these! Randomization Condition Randomization Condition: Were the data collected with suitable randomization (representative random samples or a randomized experiment)? 10% Condition 10% Condition: Is each sample size less than 10% of the population size? Nearly Normal Condition: Nearly Normal Condition: This must be checked for both groups. A violation by either one violates the condition. Independent Groups Assumption: Independent Groups Assumption: The two groups we are comparing must be independent of each other. STEPS FOR TWO-SAMPLE T- INTERVAL

9. STEPS FOR TWO-SAMPLE T- INTERVAL (CONT.)

10. Does increasing the amount of calcium in our diet reduce blood pressure? Examination of a large sample of people revealed a relationship between calcium intake and blood pressure. The relationship was strongest for black men. Such observational studies do not establish causation. Researchers therefore designed a randomized comparative experiment. The subjects in part of the experiment were 21 healthy black men. A randomly chosen group of 10 of the men received a calcium supplement for 12 weeks. The control group of 11 men received a placebo pill that looked identical. The experiment was double-blind. The response variable is the decrease in systolic (heart contracted) blood pressure for a subject after 12 weeks, in mm of mercury. An increase appears as a negative response. Take Group 1 to be the calcium group and Group 2 the placebo group. Here are the data for… The 10 men in Group 1 (calcium): 7-41817-3-511011-2 The 11 men in Group 2 (placebo): -112-1-33-552-11-1 -3 TWO-SAMPLE T-INTERVAL EXAMPLE

11. Set up your hypotheses in words and symbols as if we were to conduct a test. Check conditions Create a 90% confidence interval. Can we reject H o ? TWO-SAMPLE T-INTERVAL EXAMPLE (CONT.)

12. Check your last confidence interval with the calculator: STAT  Tests 0: 2-SampTint Specify if you are using data (if you have raw data to enter into L1 and L2) or stats (if you have summarized data) Specify 1 for both frequencies Pooled? We will discuss this later (for now say NO). Calculate TWO-SAMPLE T-INTERVAL CALCULATOR TIPS

13. two-sample t-test for means The hypothesis test we use is the two-sample t-test for means. The conditions for the two-sample t-test for the difference between the means of two independent groups are the same as for the two- sample t-interval. TWO-SAMPLE T-TESTS

14. TWO-SAMPLE T-TESTS (CONT.)

16. There are different ways to write the null hypotheses. You may often times assume they are equal: H 0 :  1 –  2 = 0ORH 0 :  1 =  2 Thus, the corresponding alternative hypotheses may be one of the following: H A :  1 –  2 ≠ 0OR H A :  1 ≠  2 H A :  1 –  2 > 0OR H A :  1 >  2 H A :  1 –  2 < 0OR H A :  1 <  2 WRITING HYPOTHESES

17. 1.Check Conditions and show that you have checked these! Randomization Condition Randomization Condition: Were the data collected with suitable randomization (representative random samples or a randomized experiment)? 10% Condition 10% Condition: Is each sample size less than 10% of the population size? Nearly Normal Condition: Nearly Normal Condition: This must be checked for both groups. A violation by either one violates the condition. Independent Groups Assumption: Independent Groups Assumption: The two groups we are comparing must be independent of each other. STEPS FOR TWO-SAMPLE MEAN HYPOTHESIS TESTING

18. STEPS FOR HYPOTHESIS TESTING (CONT.)

19. 6.Make your conclusion. STEPS FOR HYPOTHESIS TESTING (CONT.)

20. Using the previous data to compare the calcium supplement and the placebo, calculate the test statistic and P-value to determine if there is enough evidence that calcium reduces blood pressure. TWO-SAMPLE MEAN HYPOTHESIS TEST EXAMPLE

21. CALCULATOR TIPS

22. Day 1: 8.1-8.2 Book Page # 2 – 6 EVEN, 7, 9 Day 2: 8.1-8.2 Book Page # 10, 11, 19, 25, 36 ASSIGNMENTS