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

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COMPARING TWO MEANS

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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.)

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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

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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.)

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TWO-SAMPLE T-INTERVAL

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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

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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

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

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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

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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.)

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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

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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

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TWO-SAMPLE T-TESTS (CONT.)

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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

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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

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STEPS FOR HYPOTHESIS TESTING (CONT.)

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

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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

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CALCULATOR TIPS

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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

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