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CHAPTER 24: HYPOTHESES TESTS FOR TWO-SAMPLE MEANS Objective: To test claims about inferences for two sample means, under specific conditions COMPARING TWO MEANS • When comparing two means, our parameter of interest is the difference between the two means, 1 – 2. • The catch is that we must be confident that our two samples are completely independent from one anther. • Therefore, our standard deviation can be found by adding the two variances to account for extra variability among the two different samples: ( - ) = + • We still don’t know the true standard deviations of the two groups, so we need to estimate and use the standard error ( - ) = + COMPARING TWO MEANS (CONT.) • 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. • The confidence interval we build is called a two-sample t-interval (for the difference in means). • The corresponding hypothesis test is called a two-sample t-test. TWO-SAMPLE T-INTERVAL AND TTEST CONDITIONS • Independence Assumption (Each condition needs to be checked for both groups.): • Randomization Condition: Were the data collected with suitable randomization (representative random samples or a randomized experiment)? • 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 TTEST CONDITIONS (CONT.) • Normal Population Assumption: • Nearly Normal Condition: This must be checked for both groups. A violation by either one violates the condition. • 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 • When the conditions are met, we are ready to find the confidence interval for the difference between means of two independent groups, 1 – 2. The confidence interval is ( − ) ± ∗ × ( − ) where the standard error of the difference of the means is ( - ) = + • The critical value t*df depends on the particular confidence level, C, that you specify and on the number of degrees of freedom, which we get from the sample sizes and a special formula. DEGREES OF FREEDOM • The special formula for the degrees of freedom for our t critical value is a bear: 2 s12 s22 n1 n2 df 2 2 2 2 1 s1 1 s2 n1 1 n1 n2 1 n2 • Because of this, we will let technology calculate degrees of freedom for us! STEPS FOR TWO-SAMPLE TINTERVAL 1. Check Conditions and show that you have checked these! • Randomization Condition: Were the data collected with suitable randomization (representative random samples or a randomized experiment)? • 10% Condition: Is each sample size less than 10% of the population size? • Nearly Normal Condition: This must be checked for both groups. A violation by either one violates the condition. • Independent Groups Assumption: The two groups we are comparing must be independent of each other. STEPS FOR TWO-SAMPLE TINTERVAL (CONT.) 2. State the test you are about to conduct Ex) Two-Sample t-Interval for Means 4. Calculate your t-interval ∗ ( − ) ± × + 5. State your conclusion IN CONTEXT. I am 95% confident that the average pulse rate for smokers is between 3.1 and 8.9 bpm higher than the pulse rate for nonsmokers. TWO-SAMPLE T-INTERVAL EXAMPLE 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 -4 18 17 -3 -5 1 10 11 -2 3 -5 5 2 -11 -1 The 11 men in Group 2 (placebo): -1 -3 12 -1 -3 TWO-SAMPLE T-INTERVAL EXAMPLE (CONT.) • 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 CALCULATOR TIPS 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-TESTS • 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 twosample t-interval. TWO-SAMPLE T-TESTS (CONT.) • We test the hypothesis H 0:1 – 2 = 0, where the hypothesized difference, 0, is almost always 0, using the statistic − − ( − ) = ( − ) • The standard error is ( - ) = + • When the conditions are met and the null hypothesis is true, this statistic can be closely modeled by a Student’s tmodel with a number of degrees of freedom given by a TWO-SAMPLE T-TESTS (CONT.) • We test the hypothesis H 0:1 – 2 = 0, where the hypothesized difference, 0, is almost always 0, using the statistic − − ( − ) = ( − ) • The standard error is ( - ) = + • When the conditions are met and the null hypothesis is true, this statistic can be closely modeled by a Student’s tmodel with a number of degrees of freedom given by a WRITING HYPOTHESES There are different ways to write the null hypotheses. You may often times assume they are equal: • H 0: 1 – 2 = 0 OR H 0: 1 = 2 Thus, the corresponding alternative hypotheses may be one of the following: • H A: 1 – 2 ≠ 0 • H A: 1 – 2 > 0 • H A: 1 – 2 < 0 OR OR OR H A: 1 ≠ 2 H A: 1 > 2 H A: 1 < 2 STEPS FOR TWO-SAMPLE MEAN HYPOTHESIS TESTING 1. Check Conditions and show that you have checked these! • Randomization Condition: Were the data collected with suitable randomization (representative random samples or a randomized experiment)? • 10% Condition: Is each sample size less than 10% of the population size? • Nearly Normal Condition: This must be checked for both groups. A violation by either one violates the condition. • Independent Groups Assumption: The two groups we are comparing must be independent of each other. STEPS FOR HYPOTHESIS TESTING (CONT.) 2. State the test you are about to conduct Ex) Two-Sample t-Test for Means 3. Set up your hypotheses H 0: H A: 4. Calculate your test statistic = − − ( − ) + 5. Draw a picture of your desired area under the t-model, and calculate your P-value. STEPS FOR HYPOTHESIS TESTING (CONT.) 6. Make your conclusion. P-Value Action Conclusion Low Reject H0 The sample mean is sufficient evidence to conclude HA in context. High Fail to reject H0 The sample mean does not provide us with sufficient evidence to conclude HA in context. TWO-SAMPLE MEAN HYPOTHESIS TEST EXAMPLE 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. CALCULATOR TIPS Given a set of data: • Enter data into L1 • Set up STATPLOT to create a histogram to check the nearly Normal condition • STAT TESTS 2:T-Test • Choose Stored Data, then specify your data list (usually L1) • Enter the mean of the null model and indicate where the data are (>, <, or ≠) Given sample mean and standard deviation: • STAT TESTS 2:T-Test • Choose Stats enter • Specify the hypothesized mean and sample statistics • Specify the tail (>, <, or ≠) • Calculate ASSIGNMENTS: PP. 579 – 586 • Day 1: # 2 – 6 EVEN, 7, 9 • Day 2: # 10, 11, 19, 25 • Day 3: # 12, 13, 36