1. Inferences About Means from Two Groups
Two-Sample Z-Test
Inferences About Means from Two Groups
Each slide has its own narration in an audio file. For the explanation of any slide click on the audio icon to start it.
Professor Friedman's Statistics Course by H & L Friedman is licensed under a  Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 

2. Two-Sample Hypothesis Test
Researchers often want to compare two populations, with regard to a particular parameter (say, µ). For example, a researcher might wish to determine whether or not men and women are different when it comes to overall GPA in college.
Two Sample Z Test

3. Two-Sample Hypothesis Test
We test the hypothesis that there is really no difference between the two population means.
H0: μ1= μ2
H1: μ1≠ μ2
Here H0 is implying that the two populations that the samples were taken from are, in effect, a single population since there is no difference between them.
As we know, the null hypothesis, H0 is a “straw man.” We set it up and then see whether or not we can knock it down based on the sample evidence.
Two Sample Z Test

4. Two-Sample Hypothesis Test
The null hypothesis that μ1= μ2 is equivalent to stating that the difference between the two population means is 0. So H0 could be stated as: (μ1-μ2) = 0.
Note that H0 is always about population parameters, in this case the difference between the two population means.
The random variable for this test is (X̅1-X̅2), and, as always, takes its value from the sample data
Two Sample Z Test

5. Sampling Distribution of (X̅1-X̅2)
If the samples are large, random, and independent, then (X̅1-X̅2), which is a random variable, has approximately a normal distribution, with
Two Sample Z Test

6. Two-Sample Z-test
If the sample sizes are large enough, or if population standard deviations are known, we use a Two-Sample Z-test.
Two Sample Z Test

7. To Calculate Z
But, of course, under H0, µ1= µ2, so for known σ1 and σ2
Two Sample Z Test

8. To Calculate Z (cont’d)
If σ1 and σ2 are unknown, we can use Z=
as long as n1+n2 is large enough.
Two Sample Z Test

9. CIE for Difference of Means
The confidence interval estimator (CIE) for a two-sample test is used to estimate the difference between the two population means (μ1–μ2). As always, sample data can be expected to have some sampling error (also known as the margin of error). To construct a (1-)% CIE for the difference, use this formula.
The Z value in the formula comes from the Z table. We will be doing two-sided confidence intervals. Thus, for 95% CIE, use a Z of 1.96; for 90% CIE, use a Z of 1.645; for 99% CIE, use Z of 2.575; and so on.
Two Sample Z Test

10. CIE for Difference of Means(cont’d)
If the CIE includes 0, this means that we have to be willing to accept that there is a difference of 0 (fancy way of saying no difference) between the two population means).
There will always be a 0 in the CIE if we find that the confidence interval goes from a negative number to a positive number or a positive number to a negative number.
Two Sample Z Test

11. Problem 1: A New Drug
Suppose a medical researcher wants to test a new drug that is supposed to protect against the common cold.
The researcher randomly assigns subjects to one of two groups – patients in one group are given the drug; the others (the control group) take a placebo.
The measure of interest is the number of colds subjects get in a year. (Typically, most people get about 4 to 6 colds a year.)
In a “double blind” study, neither the subjects nor the researchers know who is taking the drug and who is taking the placebo.
Two Sample Z Test

12. Problem 1: A New Drug Group 1(Drug Group) Group 2 (Placebo Group)
Sample size: n1=81
Sample mean: X̅1 = 4.4 colds per year
Sample standard deviation: s1 = 0.7 colds per year
Group 2 (Placebo Group)
Sample size: n2=64
Sample mean: X̅2 = 4.8 colds per year
Sample standard deviation: s2 = 0.8 colds per year
The difference between the two sample means is -0.4 colds (4.4 – 4.8). This difference could be a statistically significant difference or could just be chance. A chance difference means that another researcher comparing two other groups might find that the placebo group has fewer colds, on average. This is why we need a statistical test.
Two Sample Z Test

13. Problem 1: A New Drug (a) Test at =.05
Therefore, REJECT H0 p < The two groups are indeed statistically different. The Drug group has fewer colds per year.
Two Sample Z Test

14. Problem 1: A New Drug
(b) Construct a 95% Confidence Interval Estimator (CIE) for the difference between the two population means.
The CIE in a two-sample situation is an estimator for the difference. We observed a difference in the data of -.4 colds. This was the difference between the two sample means. We want to construct a 95% CIE for the difference between the two population means.
If we see a 0 in the confidence interval, this means that we have to be willing to accept a difference of 0, i.e., that the two means are the same.
Two Sample Z Test

15. Problem 1: A New Drug
95% CIE for the difference between the two population means. The Z-value for a 95% confidence level is 1.96.
  ± 1.96(.127)
-.40 ± .25 Thus, the margin of sampling error is .25 colds
The 95% CIE is: ←—→ -.15
Since 0 is not in the above interval, we see that there is a difference between the two groups (The difference is not 0). We are 95% sure that the true population mean difference between the drug group and the placebo (control) group is between .15 and .65 fewer colds.
Two Sample Z Test

16. Problem 2: Science Test Scores
Scores on a standardized science test. Is there a significant difference between men and women?
(a) Test at =.05.
Two Sample Z Test

17. Problem 2: Science Test Scores
Do not reject H0 (p > .05). There is no statistically significant difference between men and women on test scores in science.
Two Sample Z Test

18. Problem 2: Science Test Scores
(b) Construct a 95% Confidence Interval Estimate for the difference between the two population means. The Z-value we use for 95% confidence is
  3.5± 1.96(2.24)
3.5± 4.4 The margin of sampling error is 4.4
The 95% CIE is: -.9 ←—→ +7.9
Since 0 is in the above interval, we conclude that there is no statistically significant difference between men and women on the science test.
Two Sample Z Test

19. Homework Practice, practice, practice.
Do lots and lots of problems. You can find these in the online lecture notes.
Two Sample Z Test