Independent Samples Comparing Means Lecture 37 Sections 11 1 11 2 11 4 Fri Apr 6 2007 The t Distribution Whenever 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 sp n1 1 s12 n2 1 s2 2 n1 n2 2 x1 x2 and the t Distribution If 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 t x1 x2 1 sp 1 1 n1 n2 2 with df n1 n2 2 Hypothesis Testing See Example 11 4 p 699 Comparing Two Headache Treatments State the hypotheses H0 1 2 H1 1 2 State the level of significance 0 05 The t Statistic Compute the value of the test statistic The test statistic is t x1 x2 1 1 sp n1 n2 with df n1 n2 2 Computations 2 2 9 s1 9 s2 sp 5 052 18 22 6 19 4 t 1 416 1 1 5 052 10 10 Hypothesis Testing Calculate the p value The number of degrees of freedom is df df1 df2 18 p value P t 1 416 tcdf 1 416 E99 18 0 0869 Hypothesis Testing State the decision Accept H0 State the conclusion Treatment 1 is more effective than Treatment 2 The TI 83 and Means of Independent Samples Stats Press STAT TESTS Choose 2 SampTTest Choose Stats The TI 83 and Means of Independent Samples Stats Provide 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 Stats Select Calculate and press ENTER The display shows among other things the value of the test statistic and the pvalue The TI 83 and Means of Independent Samples Data Enter the data from the first sample into L 1 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 Data Provide 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 Data Select Calculate and press ENTER The display shows among other things the value of the test statistic and the pvalue Paired vs Independent Samples The following data represent students calculus test scores before and after taking an algebra refresher course Student 1 2 3 4 5 6 7 8 Before 85 63 94 78 75 82 45 58 After 92 68 98 83 80 88 53 62 Paired vs Independent Samples Perform a test of the hypotheses H0 2 1 0 H1 2 1 0 treating the samples as independent Paired vs Independent Samples Had we performed a test of the same hypotheses H0 D 0 H1 D 0 treating the samples as paired then the pvalue would have been 0 000005688 Why so small Paired Samples Why is there a difference 100 90 80 70 60 50 40 1 2 3 4 5 Paired 6 7 8 Independent Samples Why is there a difference 100 90 80 70 60 50 40 1 2 3 4 5 Independent 6 7 8 Confidence Intervals Confidence intervals for 1 2 use the same theory The point estimate is x1 x2 The standard deviation of x1 x2 is approximately sp 1 1 n1 n2 Confidence Intervals The confidence interval is 1 1 known large samples x1 x2 z n1 n2 or x1 x2 z s p or x1 x2 t s p 1 1 unknown large samples n1 n2 1 1 n1 n2 unknown normal pops small samples Confidence Intervals The choice depends on is known Whether the populations are normal Whether the sample sizes are large Whether Example Find 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 is 3 2 2 101 2 259 3 2 4 75 The TI 83 and Means of Independent Samples To 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 yes
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