Independent Samples: Comparing MeansThe t DistributionEstimating x1 –x2 and the t DistributionHypothesis TestingThe t StatisticComputationsSlide 8Slide 9The TI-83 and Means of Independent Samples – StatsSlide 11Slide 12The TI-83 and Means of Independent Samples – DataSlide 14Slide 15Paired vs. Independent SamplesSlide 17Slide 18Paired SamplesIndependent SamplesConfidence IntervalsSlide 22Slide 23ExampleThe TI-83 and Means of Independent SamplesIndependent Samples: Comparing MeansLecture 37Sections 11.1 – 11.2, 11.4Fri, Apr 6, 2007The t DistributionWhenever we do not know , we must use s1 and s2 to estimate .In this case, we will have to use the t distribution instead of the standard normal distribution, unless the sample sizes are large.Estimating Individually, s1 and s2 estimate .However, we can get a better estimate than either one if we “pool” them together.The pooled estimate is .21121222211nnsnsnspx1 –x2 and the t DistributionIf we use sp instead of , and the sample sizes are small, then we should use t instead of Z.The number of degrees of freedom is df = df1 + df2 = n1 + n2 – 2.That is 2 with ,1121212121 nndfnnsxxtpHypothesis TestingSee Example 11.4, p. 699 – Comparing Two Headache Treatments.State the hypotheses.H0: 1 = 2H1: 1 > 2State the level of significance. = 0.05.The t StatisticCompute the value of the test statistic.The test statistic iswith df = n1 + n2 – 2.212111nnsxxtpComputations.416.1101101052.54.196.22.052.518992221tssspHypothesis TestingCalculate the p-value.The number of degrees of freedom isdf = df1 + df2 = 18.p-value = P(t > 1.416)= tcdf(1.416, E99, 18)= 0.0869.Hypothesis TestingState the decision.Accept H0.State the conclusion.Treatment 1 is more effective than Treatment 2.The TI-83 and Means of Independent Samples – StatsPress STAT > TESTS.Choose 2-SampTTest.Choose Stats.The TI-83 and Means of Independent Samples – StatsProvide the information that is called for.x1, s1, n1.x2, s2, n2.Alternative hypothesis.Whether to use a pooled estimate of . Answer “yes.”The TI-83 and Means of Independent Samples – StatsSelect Calculate and press ENTER.The display shows, among other things, the value of the test statistic and the p-value.The TI-83 and Means of Independent Samples – DataEnter the data from the first sample into L1.Enter the data from the second sample into L2.Press STAT > TESTS.Choose 2-SampTTest.Choose Data.The TI-83 and Means of Independent Samples – DataProvide the information that is called for.List 1: L1.List 2: L2.Freq 1: 1.Freq 2: 1.Alternative hypothesis.Whether to use a pooled estimate of . Answer “yes.”The TI-83 and Means of Independent Samples – DataSelect Calculate and press ENTER.The display shows, among other things, the value of the test statistic and the p-value.Paired vs. Independent SamplesThe following data represent students’ calculus test scores before and after taking an algebra refresher course.Student 1 2 3 4 5 6 7 8Before 85 63 94 78 75 82 45 58After 92 68 98 83 80 88 53 62Paired vs. Independent SamplesPerform a test of the hypothesesH0: 2 – 1 = 0H1: 2 – 1 > 0treating the samples as independent.Paired vs. Independent SamplesHad we performed a test of the “same” hypothesesH0: D = 0H1: D > 0treating the samples as paired, then the p-value would have been 0.000005688.Why so small?Paired SamplesWhy is there a difference?1 2 3 54 6 87405060809010070PairedIndependent SamplesWhy is there a difference?1 2 3 54 6 87405060809010070IndependentConfidence IntervalsConfidence intervals for 1 – 2 use the same theory.The point estimate isx1 –x2.The standard deviation ofx1 –x2 is approximately2111nnspConfidence IntervalsThe confidence interval isoror212111nnzxx212111nnszxxp212111nnstxxp( known, large samples)( unknown, large samples)( unknown, normal pops.,small samples)Confidence IntervalsThe choice depends onWhether is known.Whether the populations are normal.Whether the sample sizes are large.ExampleFind a 95% confidence interval for 1 – 2 in Example 11.4, p. 699.x1 –x2 = 3.2.sp = 5.052.Use t = 2.101.The confidence interval is3.2 (2.101)(2.259) = 3.2 4.75.The TI-83 and Means of Independent SamplesTo find a confidence interval for the difference between means on the TI-83,Press STAT > TESTS.Choose either 2-SampZInt or 2-SampTInt.Choose Data or Stats.Provide the information that is called for.2-SampTTest will ask whether to use a pooled estimate of . Answer
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