1. Two Sample Statistical Inferences
Additional material

2. How can we compare two means?
Suppose that we have two populations. Furthermore, suppose that the first population has a mean 1 and the second population a mean 2.
We would like to use inferential statistics to compare 1 and 2 by using either a confidence interval and/or a significance test.

3. Independent Samples
Independent Samples occur when observations are gathered independently from two different populations.
For example, suppose we want to compare starting salaries for males and females at a certain bank. We collect a random sample of males and a random sample of females and measure their starting salaries.

4. Paired Samples
Paired Samples occur when observations are gathered in pairs, where each pair consists of an observation from population 1 and an observation from population 2, and these two observations are related.
Although there are other types of paired samples, a common type is a before-after sample.

5. Paired Samples
For example, suppose we want to determine the effectiveness of a new drug in reducing blood pressure.
For each individual in the sample a measurement is taken prior to the medication and another measurement is taken after using the medication for a fixed period of time.
These two observations (for each individual) are related. This constitutes a paired sample.

6. Outline Two sample comparisons:
Paired Samples (dependent samples)
Independent Samples – Population Standard Deviations Not Assumed to be Equal (σ1≠ σ2)
Independent Samples – Population Standard Deviations Assumed to be Equal (σ1= σ2)
All methods assume random samples from normal populations with unknown means and unknown standard deviations.
For small sample sizes, it is critical that our observations are normal or approximately normal.
However, for sufficiently large sample sizes (from the C.L.T.) the assumptions of normality can be relaxed. 6 6

7. Paired Samples 7 7

8. Paired (dependent) Samples

9. Paired (dependent) Samples

10. Paired (dependent) Samples
t-score = di invttail(n-1,α/2)

11. Paired (dependent) Samples

12. Independent Two Samples12 12

13. Independent Two Samples
We gather independent and random samples from each population. 13 13

14. Independent Two Samples14 14

15. Independent Two Samples15 15

16. Independent Two Samples: Not Assuming Equal Standard Deviations (σ1≠ σ2)
~ tdf=v,
Note: Unless we obtain our degrees of freedom using a computer, we will round v down to the nearest integer. 16 16

17. Independent Two Samples: Not Assuming Equal Standard Deviations (σ1≠ σ2)
A (1-α)×100% CI is
where a critical value that makes the right tail probability equal to α/2
Stata: di invttail(df, α/2) 17 17

18. Independent Two Samples: Not Assuming Equal Standard Deviations (σ1≠ σ2)18 18

19. Exercise: Does alcohol affect males and females differently?
A study involving males and females with similar physical characteristics was conducted. In a controlled setting, each individual was asked to consume 4 ounces of alcohol. One hour after consumption each participant took a Breathalyzer to measure his or her blood alcohol level. The following results were obtained:
Is there a real difference in blood alcohol levels between males and females? Carry out an appropriate test to answer this question.
Calculate a 99% confidence interval for the difference in blood alcohol levels between males and females. 19 19

20. Exercise Stata command and outputs:
ttesti n1 u1 s1 n2 u2 s2, unequal level(#) 20

21. Independent Samples: Assuming Equal Standard Deviations (σ1= σ2)21 21

22. Independent Samples: Assuming Equal Standard Deviations (σ1= σ2)22 22

23. Independent Samples: Assuming Equal Standard Deviations (σ1= σ2)23 23

24. Independent Samples: Assuming Equal Standard Deviations (σ1= σ2)24 24

25. To be pooled, or not to be pooled? Equal S.D. or not equal?
We could base our decision on an informal rule (usually works and is much less complicated).
If no sample standard deviation is twice the other, (i.e 0.5 < s1/s2 < 2), then the assumption of equal standard deviations should be ok.
We could perform a graphical analysis and look at the box-plots for the samples to informally assess the equal standard deviations assumption.
There are formal tests to assess the evidence against equal population standard deviations
Variance Ratio Test
Levene’s Test 25 25

26. Variance Ratio Test Ho: σ12 = σ22 vs. Ha: σ12 ≠ σ22
Test statistic: F= s12 / s22 ~ Fdf1=n1-1, df2=n2-1
p-value: 26

27. Variance Ratio Test Stata command: sdtesti n1 . sd1 n2 . sd2
Example: Test σ1= σ2 when observed n1=75, sd1=6.5, n2=65, and sd2= 7.5
sdtesti 27

28. Exercise
Two machines are used to fill plastic bottles with dishwashing detergent. Two random samples are taken and the following results are obtained:
Is there a real difference between their average fills? Carry out an appropriate test to answer this question.
Calculate a 95% confidence interval for the difference in the mean fill between the two machines 28 28

29. Exercise
s1=√0.113=0.336; s2= √0.125 =0.354
Testing for Equal Variances 29

30. Exercise Stata command for two-sample t test with equal variances
ttesti N1 u1 s1 N2 u2 S2, level(#)
Compared with machine 2, machine 1 had a statistically significantly lower mean fill. The difference in mean fill was (95%CI: to -0.23) for machine 2 relative to machine 1 (p<0.001). 30