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Section 10.2 – Significance Tests for the Difference of Two MeansConditions for Significance Tests for Two Means are the same as they were for a Confidence Interval for the difference of Two Means.Significance Tests for 2-meansH0: μ1=μ2 μ1-μ2=0Two-Sample t Test for the Difference between Two MeansSuppose the conditions are met. To test the hypothesis H0: μ1-μ2=0 , compute the t-statistic t=x1-x2-0s12n1+s22n2 Find the P-value by calculating the probability of getting a t-statistic this large or larger in the direction specified by the alternative hypothesis Ha. Use the t distribution with degrees of freedom approximated by technology or the smaller of n1 – 1 and n2 – 1. Example, p. 645 - 646: 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 were 21 healthy black men who volunteered to take part in the experiment. They were randomly assigned to two groups: 10 of the men received a calcium supplement for 12 weeks, while the control group of 11 m3n received a placebo pill that looked identical. The experiment was double-blind. The response variable is the decrease in systolic (top number) blood pressure for a subject after 12 weeks, in millimeters of mercury. An increase appears as a negative number. Here are the data:Group 1 (calcium):7-41817-3-511011-2Group 2 (placebo):-112-1-33-552-11-1-3Do the data provide convincing evidence that a calcium supplement reduces blood pressure more than a placebo? Carry out an appropriate test to support your answer.State: H0: μ1=μ2 μ1-μ2=0 Represents that the mean stayed the sameHa: μ1>μ2 μ1-μ2>0 Represents the calcium causing a greater decrease in b.p.?1 = true mean decrease in systolic blood pressure for healthy black men like the ones in this study who take a calcium supplement?2 = true mean decrease in systolic blood pressure for healthy black men like the ones in this study who take a placeboNo significance level was specified, so we’ll use α = 0.05.Plan: Two – sample t-test for μ1-μ2 2 population means, so 2-sample. t-test since we have s and not σ.Conditions: Random: 21 subjects were randomly assigned to the two treatment groups.35191701187450010%: Don’t need to check because there was no sampling. Normal/Large Sample: Plot Data1498763152400No strong skewness or outliers in the boxplots, so it is safe to use t procedures.020000No strong skewness or outliers in the boxplots, so it is safe to use t procedures.Do: Data: (You can find this by entering data in your calculator and using 2 variable stats.)nx Mean change in b.p.s1Calcium105.0008.7432Placebo11-0.2735.901 x1-x2=5--.273=5.273t=x1-x2-μ1-μ2s12n1+s22n2=5.273-08.743210+5.901211≈1.604*In table, use smallest degrees of freedomn1-1=9 lies between 1.383 and 1.833, so p-value between 0.05 and 0.10n2-1=10Calculator: STAT TESTS 2-SampTTest Highlight Stats and type in your values. (OR YOU CAN USE DATA SINCE WE HAVE THE DATA)P-value = 0.0644Conclude:Since the P-value of 0.0644 is greater than 0.05, we fail to reject H0. We do not have convincing evidence that the true mean decrease in systolic blood pressure is higher for men like these who take calcium than for men like these who take a placebo. ................
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