1. Comparing Two Populations
Section
Lesson11.2.1

2. Starter
Write the assumptions that underlie the use of the t procedures.
Which assumption is so important you can’t work without it?
What is the best way to tell if the distribution assumption is met?

3. Starter Solution The two main assumptions we need are:
The sample came from a valid SRS of the population.
The population is approximately normally distributed.
We can’t do anything without a valid SRS.
Plot the data to see if they are approximately normal.
Note that if sample size is large we can live with skewness or outliers.

4. Today’s Objectives
Students will compare the means of two populations to determine whether the differences are statistically significant.
Students will perform a two-sample t-test from formula.
Students will form a two-population confidence interval from formula.
California Standard 20.0
Students are familiar with the t distributions and t test and understand their uses.

5. Two-sample Problems
Used when we want to compare two different populations or experimental treatments
This is how we will compare the placebo group with the treatment group
We need a separate sample from each population
That’s why it’s called “two-sample”
The goal is to compare the population means to determine if they are different (in a statistically significant way)
This is not matched pairs: we don’t have information about individual scores

6. Assumptions for comparing two means
There are two SRS’s from distinct populations
Both populations are normally distributed.
We do not know μ and σ.
With large sample sizes, the Central Limit Theorem assures approximately normal sample means.
The samples are independent
One sample has no influence on the other.
Matching violates independence.

7. Procedure Assumptions Assumptions So Far… One population t for means
Two population t for means
Assumptions
SRS, Normal Dist
SRS, Normal, Independent

8. Population symbols we will use
We want to compare μ1 and μ2
Are they equal or different?
To do compare, we use sample means.
Population
Variable
Mean
Std Dev 1 x1 μ1 σ1 2 x2 μ2 σ2

9. Sample symbols we will use
We will use sample means Ë1 and Ë2 to draw inferences about population means μ1 and μ2
Population
Sample
Size
Mean
Std Dev 1 n1 Ë1 s1 2 n2 Ë2 s2

10. The distribution of Ë1 – Ë2
Shape: The distribution of all possible differences (Ë1 - Ë2) is approximately normal
Center: The mean of the distribution of all possible differences (Ë1 – Ë2) is (μ1 – μ2 )
Spread: The standard error of the distribution is given by the formula

11. Two-sample Hypothesis Tests
Establish hypotheses.
Ho: μ1 = μ2
Ha: μ1 ≠ μ2 OR µ1 < μ2 OR µ1 > μ2
Calculate the t-test statistic.
Formula on next slide
Find the p-value from table or tcdf command.
Write a three-phrase conclusion statement.

12. The two-sample t statistic
Note: The degrees of freedom will be n-1 for the smallest sample size.

13. Example 11.9
A test of social abilities is given to 133 men and 162 women. Here are the results
Is there a significant difference between the two populations (men and women)?
Group
Sex n Ë s 1
Male
133
25.34
5.05 2
Female
162
24.94
5.44

14. State the hypotheses Calculate the t statistic
Ho: μ1 = μ2 Ha: μ1 ≠ μ2
Now apply the formula for t.
You should have gotten t = 0.654
So what is P?
tcdf(.654, 999, 132) = 0.257
(Remember d.f. = smallest n-1)
But this is a two-tailed test, so:
P = 2 x = 0.514
Conclusion: Because p is greater than any reasonable alpha, there is not enough evidence to support the claim that men and women’s scores differ.

15. Two-sample Confidence Interval
As before, confidence intervals take the form
Estimate ± critical value x standard error
So the confidence interval for μ1 – μ2 is

16. Example 11.9 again
Calculate a 95% Confidence Interval for the mean difference between men and women on the social skills test

17. Example 11.9 again
Calculate a 95% Confidence Interval for the mean difference between men and women on the social skills test

18. Example 11.9 again
Calculate a 95% Confidence Interval for the mean difference between men and women on the social skills test.
Notice that the interval contains zero, so this is consistent with the failure to reject in the hypothesis test.

19. Today’s Objectives
Students will compare the means of two populations to determine whether the differences are statistically significant.
Students will perform a two-sample t-test from formula.
Students will form a two-population confidence interval from formula.
California Standard 20.0
Students are familiar with the t distributions and t test and understand their uses.

20. Homework
Read pages 617 – 627
Do problems