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Two-Sample Problems<br />The goal of this type of inference <br />compare the responses of two treatments -or-<br />compare the characteristics of two populations<br />Separate samples from each population<br />Responses of each group are independent of those in the other group<br />
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Before We Begin<br />This is another set of PHANTOMS procedures<br />It is important to note that “two populations” means that there is no overlap in the samples<br />The sample sizes do not need to be equal<br />
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Hypotheses<br />There are two styles of writing hypotheses<br />Style 1<br />H0: 1 = 2<br />Ha: 1 2, or<br />Ha: 1 > 2, or<br />Ha: 1 < 2<br />
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Hypotheses<br />There are two styles of writing hypotheses<br />Style 2<br />H0: 1 - 2 = 0<br />Ha: 1 - 2 0, or<br />Ha: 1 - 2 > 0 (this implies 1 > 2), or<br />Ha: 1 - 2 < 0 (this implies 1 < 2)<br />This style is more versatile<br />since it allows you to use adifference other than zero<br />
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Assumptions<br />Simple Random Sample<br />Each sample must be from an SRS<br />Independence<br />Samples may not influence each otherNo paired data!N1 > 10n1and N2 > 10n2<br />(if sampling w/o replacement)<br />
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Assumptions<br />Normality (of sampling distibution)<br />large samples (n1 > 30 and n2 > 30)this is the Central Limit Theorem -OR-<br />medium samples (15<n1<30 and 15<n2<30)-Histogram symmetric or slight skew and single peak-Norm prob plots for n1 and n2 are linear-No Outliers -OR-<br />
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Assumptions<br />Normality (of sampling distibution)<br />small samples (n1<15 and n2<15)-Histogram symmetric and single peak-Norm prob plots for n1 and n2 are linear-No Outliers<br />
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2-sample test statistic<br />z-tests<br />t-tests<br />df = smaller of<br /> n1 -1 or n2 - 1<br />
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Example 13.2<br />Researchers designed a randomized comparative experiment to establish the relationship between calcium intake and blood pressure in black men. Group 1 (n1 = 10) took calcium supplement, Group 2 (n2 =11) took a placebo. The response is the decrease in systolic blood pressure<br />Group 1: 7, -4, 18, 17, -3, -5, 1, 10, 11, -2<br />Group 2: -1, 12, -1, -3, 3, -5, 5, 2, -11, -1, -3<br />
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Example 13.2<br />Parameter<br /> 1 - 2 = difference in average systolic blood pressure in healthy black men between the calcium regimen and the placebo regimen <br /> xbar1 - xbar2 = difference in average systolic blood pressure in healthy black men in the two samples between the calcium regimen and the placebo regimen <br />
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Example 13.2<br />Assumptions<br />Simple Random Sample<br />We are told that both samples come from a randomized design<br />Independence<br />Both samples are independent, and (n1) N1 > 10(10) =100, (n2) N2 > 10(11)=110the population of black men is greater than 110<br />
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Example 13.2<br />Assumptions (cont)<br />Sample 1Sample 2<br />
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Example 13.2<br />Assumptions (cont)<br />Normality<br />Both samples are single peaked with moderate skewness and approximately normal with no outliers.<br />Although sample 1 shows some skewness, the t-procedures are robust enough to handle this skew.<br />
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Example 13.2<br />Name of Test<br />We will conduct a 2-sample t-test for population means<br />Test Statistic<br />
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Example 13.2<br />P Value<br />Decision<br />Fail to Reject H0 at the 5% significance level<br />
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Example 13.2<br />Summary<br />Approximately 7% of the time, our samples of size 10 and 11 would produce a difference at least as extreme as 5.2727<br />Since this p-value is not less than the presumed = 0.05, we will fail to reject H0 <br />We do not have enough evidence to conclude that calcium intake reduces the average blood pressure in healthy black men. <br />
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Confidence Intervals<br /> Confidence Interval for a difference to two sample means<br />
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Robustness<br /> 2-sample t-procedures are more robust than one sample procedures. They can be used for sample sizes as small as n1 = n2 = 5 when the samples have similar shapes.<br />Guidelines for using t-procedures<br />n1 + n2< 15: data must be approx normal,no outliers<br />n1 + n2 >15: data can have slight skew, no outliers<br />n1 + n2> 30: data can have skew<br />
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Degrees of Freedom<br />We have been using the smaller of n1 or n2 to determine the df<br />This will ensure that our pvalue is smaller than the calculated pvalueand confidence intervals are smaller than calculated.<br />These are “worst case scenario” calculations<br />There is a more exact df formula on p792<br />Your calculator also uses a df formula for two samples<br />You do not need to memorize these other formulas!<br />
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Calculators<br />The tests we are using are located in the [STAT] -> “TESTS” menu<br />2-SampZTest = two sample z-test for means<br />2-SampTTest = two sample t-test for mean<br />2-SampZInt = two sample z Confidence Interval for difference of means<br />2-SampTInt = two sample t Confidence Interval for difference of means<br />
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Calculators<br />Freq1 and Freq2 should be set to “1”<br />Pooled should be set to “NO”<br />
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2-Sample Inference for Proportions<br />We are testing to see if<br />Two populations have the same proportion OR<br />A treatment affects the proportion <br />Remember: this is not a procedure for paired data (matched pair design/pre- and post-test)<br />
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Combined Proportion<br />One of the underlying assumptions of the test is that the two proportions actually come from the same population.<br />The test makes use of the “combined proportion” as below:<br />
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Hypotheses<br />There are two styles of writing hypotheses<br />Style 1<br />H0: p1 = p2<br />Ha: p1 p2, or<br />Ha: p1 > p2, or<br />Ha: p1 < p2<br />
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Hypotheses<br />There are two styles of writing hypotheses<br />Style 2<br />H0: p1 - p2 = 0<br />Ha: p1 - p2 0, or<br />Ha: p1 - p2 > 0 (this implies p1 > p2), or<br />Ha: p1 - p2 < 0 (this implies p1 < p2)<br />This style is more versatile<br />since it allows you to use adifference other than zero<br />
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Assumptions<br />Simple Random SampleBoth samples must be viewed as an SRS from their respective population or two groups from a randomized experiment<br />Independence<br />N1 > 10n1 and N2 > 10n2<br />Normality<br />n1(pchat)> 5, n1(qchat)> 5 and n2(pchat)> 5, n2(qchat)> 5<br />
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Test Statistic<br />The test statistic for proportions is always from the Normal distribution <br />
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Example 13.9<br /> A study was conducted to find the effects of preschool programs in poor children. Group 1 (n=61) had no preschool and group 2 (n=62) had similar backgrounds and attended preschool. The study measured the need for social services when the children became adults. After investigation it was found that p1hat = 49/61 and p2hat = 38/62.Does the data support the claim that preschool reduced the social services claimed?<br />
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Example 13.9<br />Parameters<br />p1 = proportion of adults who did not receive preschool and file for social services<br />p2 = proportion of adults who received preschool and filed for social services<br />p1hat = proportion of adults in group 1who did not receive preschool and file for social services<br />p2hat = proportion of adults in group 2 who received preschool and filed for social services<br />
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Example 13.9<br />Hypotheses<br />H0: p1 – p2 = 0<br />Ha: p1 – p2 > 0<br />The proportion of non-preschool is greater than that of pre-school<br />
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Example 13.9<br />Assumptions<br />Simple Random SampleSince the measurements are from a randomized experiment, we can assume that they are from an SRS<br />IndependenceN1 > 10(61) = 610: more than 610 do not attend preschoolN2 > 10(62) = 620: more than 620 attend preschool<br />Normality61(.70) = 42.7 > 5, 61(.30) = 18.3 > 562(.70) = 43.4 > 5, 62(.30) = 18.6 > 5<br />
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Example 13.9<br />Name of Test<br />2-Sample Z-test for proportions<br />Test Statistic<br />
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Example 13.9<br />Summary<br />Approximately 1% of the time, two samples of size 61 and 62 will produce a difference of at least 0.190.<br />Since our p value is less than an of 0.05, we will reject our H0.<br />Our evidence supports the claim that enrollment in preschool reduces the proportion of adults who file social services claims.<br />
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Confidence Intervals<br />The confidence interval for the difference between the proportions of two samples is given as:<br />Notice that the Confidence Interval does not use pchat and qchat.<br />
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Confidence Intervals<br />Assumptions<br />Simple Random SampleBoth samples must be viewed as an SRS from their respective population or two groups from a randomized experiment<br />Independence<br />N1 > 10n1 and N2 > 10n2<br />Normality<br /> n1(p1)> 5, n1(q1)> 5 and n2(p2)> 5, n2(q2)> 5(again, not pc or qc)<br />
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Calculators<br /> The tests we are using are located in the [STAT] -> “TESTS” menu<br />2-PropZTest = 2 proportion z-test<br />2-PropZInt = 2 proportion confidence interval<br />
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