3. Two-Sample Problems The goal of this type of inference compare the responses of two treatments -or- compare the characteristics of two populations Separate samples from each population Responses of each group are independent of those in the other group
4. Before We Begin This is another set of PHANTOMS procedures It is important to note that “two populations” means that there is no overlap in the samples The sample sizes do not need to be equal
5. Hypotheses There are two styles of writing hypotheses Style 1 H0: 1 = 2 Ha: 1 2, or Ha: 1 > 2, or Ha: 1 < 2
6. Hypotheses There are two styles of writing hypotheses Style 2 H0: 1 - 2 = 0 Ha: 1 - 2 0, or Ha: 1 - 2 > 0 (this implies 1 > 2), or Ha: 1 - 2 < 0 (this implies 1 < 2)
7. Hypotheses There are two styles of writing hypotheses Style 2 H0: 1 - 2 = 0 Ha: 1 - 2 0, or Ha: 1 - 2 > 0 (this implies 1 > 2), or Ha: 1 - 2 < 0 (this implies 1 < 2) This style is more versatile since it allows you to use adifference other than zero
8. Assumptions Simple Random Sample Each sample must be from an SRS Independence Samples may not influence each otherNo paired data!N1 > 10n1and N2 > 10n2 (if sampling w/o replacement)
9. Assumptions Normality (of sampling distibution) large samples (n1 > 30 and n2 > 30)this is the Central Limit Theorem -OR- 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-
10. Assumptions Normality (of sampling distibution) small samples (n1<15 and n2<15)-Histogram symmetric and single peak-Norm prob plots for n1 and n2 are linear-No Outliers
12. Example 13.2 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 Group 1: 7, -4, 18, 17, -3, -5, 1, 10, 11, -2 Group 2: -1, 12, -1, -3, 3, -5, 5, 2, -11, -1, -3
13. Example 13.2 Parameter 1 - 2 = difference in average systolic blood pressure in healthy black men between the calcium regimen and the placebo regimen xbar1 - xbar2 = difference in average systolic blood pressure in healthy black men in the two samples between the calcium regimen and the placebo regimen
15. Example 13.2 Assumptions Simple Random Sample We are told that both samples come from a randomized design Independence Both samples are independent, and (n1) N1 > 10(10) =100, (n2) N2 > 10(11)=110the population of black men is greater than 110
17. Example 13.2 Assumptions (cont) Normality Both samples are single peaked with moderate skewness and approximately normal with no outliers. Although sample 1 shows some skewness, the t-procedures are robust enough to handle this skew.
18. Example 13.2 Name of Test We will conduct a 2-sample t-test for population means Test Statistic
19. Example 13.2 P Value Decision Fail to Reject H0 at the 5% significance level
20. Example 13.2 Summary Approximately 7% of the time, our samples of size 10 and 11 would produce a difference at least as extreme as 5.2727 Since this p-value is not less than the presumed = 0.05, we will fail to reject H0 We do not have enough evidence to conclude that calcium intake reduces the average blood pressure in healthy black men.
22. Robustness 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. Guidelines for using t-procedures n1 + n2< 15: data must be approx normal,no outliers n1 + n2 >15: data can have slight skew, no outliers n1 + n2> 30: data can have skew
23. Degrees of Freedom We have been using the smaller of n1 or n2 to determine the df This will ensure that our pvalue is smaller than the calculated pvalueand confidence intervals are smaller than calculated. These are “worst case scenario” calculations There is a more exact df formula on p792 Your calculator also uses a df formula for two samples You do not need to memorize these other formulas!
24. Calculators The tests we are using are located in the [STAT] -> “TESTS” menu 2-SampZTest = two sample z-test for means 2-SampTTest = two sample t-test for mean 2-SampZInt = two sample z Confidence Interval for difference of means 2-SampTInt = two sample t Confidence Interval for difference of means
27. 2-Sample Inference for Proportions We are testing to see if Two populations have the same proportion OR A treatment affects the proportion Remember: this is not a procedure for paired data (matched pair design/pre- and post-test)
28. Combined Proportion One of the underlying assumptions of the test is that the two proportions actually come from the same population. The test makes use of the “combined proportion” as below:
29. Hypotheses There are two styles of writing hypotheses Style 1 H0: p1 = p2 Ha: p1 p2, or Ha: p1 > p2, or Ha: p1 < p2
30. Hypotheses There are two styles of writing hypotheses Style 2 H0: p1 - p2 = 0 Ha: p1 - p2 0, or Ha: p1 - p2 > 0 (this implies p1 > p2), or Ha: p1 - p2 < 0 (this implies p1 < p2)
31. Hypotheses There are two styles of writing hypotheses Style 2 H0: p1 - p2 = 0 Ha: p1 - p2 0, or Ha: p1 - p2 > 0 (this implies p1 > p2), or Ha: p1 - p2 < 0 (this implies p1 < p2) This style is more versatile since it allows you to use adifference other than zero
32. Assumptions Simple Random SampleBoth samples must be viewed as an SRS from their respective population or two groups from a randomized experiment Independence N1 > 10n1 and N2 > 10n2 Normality n1(pchat)> 5, n1(qchat)> 5 and n2(pchat)> 5, n2(qchat)> 5
33. Test Statistic The test statistic for proportions is always from the Normal distribution
34. Example 13.9 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?
35. Example 13.9 Parameters p1 = proportion of adults who did not receive preschool and file for social services p2 = proportion of adults who received preschool and filed for social services p1hat = proportion of adults in group 1who did not receive preschool and file for social services p2hat = proportion of adults in group 2 who received preschool and filed for social services
36. Example 13.9 Hypotheses H0: p1 – p2 = 0 Ha: p1 – p2 > 0 The proportion of non-preschool is greater than that of pre-school
37. Example 13.9 Assumptions Simple Random SampleSince the measurements are from a randomized experiment, we can assume that they are from an SRS IndependenceN1 > 10(61) = 610: more than 610 do not attend preschoolN2 > 10(62) = 620: more than 620 attend preschool Normality61(.70) = 42.7 > 5, 61(.30) = 18.3 > 562(.70) = 43.4 > 5, 62(.30) = 18.6 > 5
38. Example 13.9 Name of Test 2-Sample Z-test for proportions Test Statistic
40. Example 13.9 Summary Approximately 1% of the time, two samples of size 61 and 62 will produce a difference of at least 0.190. Since our p value is less than an of 0.05, we will reject our H0. Our evidence supports the claim that enrollment in preschool reduces the proportion of adults who file social services claims.
41. Confidence Intervals The confidence interval for the difference between the proportions of two samples is given as: Notice that the Confidence Interval does not use pchat and qchat.
42. Confidence Intervals Assumptions Simple Random SampleBoth samples must be viewed as an SRS from their respective population or two groups from a randomized experiment Independence N1 > 10n1 and N2 > 10n2 Normality n1(p1)> 5, n1(q1)> 5 and n2(p2)> 5, n2(q2)> 5(again, not pc or qc)
43. Calculators The tests we are using are located in the [STAT] -> “TESTS” menu 2-PropZTest = 2 proportion z-test 2-PropZInt = 2 proportion confidence interval
