1. Section 13.1
Comparing Two Means
Objectives:
1. Identify situations in which two-sample problems might arise.
2. Describe the three conditions necessary for doing inference
involving two population means.
3. Clarify the difference between the two-sample z statistic and the
two-sample t statistic.
4. Identify the two practical options for using two-sample t procedures
and how they differ in terms of computing the number of degrees of
freedom.
5. Conduct a two-sample significance test for the difference between
two independent means using the Inference Toolbox.
6. Compare the robustness of two-sample procedures with that of
one-sample procedures. Discuss the role of equal sample size.
7. Explain what is meant by "pooled two-sample t procedures," when
pooling can be justfied, and why it is advisable not to pool.
2. Inference for Two Independent Samples
Pop Pop Sample Sample
Population Variable Mean S.D. Mean S.D.
1 X1 µ1 σ1 x1 s1
2 X2 µ2 σ2 x2 s2
You should use more descriptive subscripts than "1"
and "2" when doing an actual problem.
3. Inference for Two Independent Samples
Conditions:
1) SRS from both populations
2) Both sampling distributions are normally
distributed.
3) Sample size is less than 1/10 population
size.
4) σ from at least one population is unknown.
4. Inference for Two Independent Samples
We will be doing inference primarily about the difference
between two means.
So, for a confidence interval, we will look at µ1 – µ2.
For a hypothesis test, our null hypothesis will typically be:
H0: µ1 = µ2 or H0: µ1 – µ2 = 0
5. Inference for Two Independent Samples
A statistics student designed an experiment to test the life of brand
name and generic batteries. He used 6 pairs of AA batteries from
each of a brand name and generic battery manufacturer.
He kept a battery powered CD player running with the same CD set
on the same volume until no more music was heard.
Identify each of the following:
Factor
Level
Response variable
To account for changes in CD player performance over time, he
randomized the order of the batteries.
6. Inference for Two Independent Samples
Here is the data:
Brand Name Generic
194.0 190.7
205.5 203.5
199.2 203.5
172.4 206.5
184.0 222.5
169.5 209.4
What should we do first?
7. Inference for Two Independent Samples
Always look at the data before performing any
inference procedure!!
When working with two samples, it is frequently
a good idea to do some kind of comparative plot
(back to back stem-and-leaf plots, or side-by-
side boxplots.)
8. Inference for Two Independent Samples
Why is this situation independent samples and
not matched pairs?
9. Inference for Two Independent Samples
Perform the hypothesis test, using the
inference toolbox.
10. Inference for Two Independent Samples
Wait a minute! What do we use for a test
statistic?
Recall that the general form for any test statistic
is:
11. Inference for Two Independent Samples
So, our test statistic will have the form:
12. Inference for Two Independent Samples
Terminology Note:
Standard Deviation of the Sample Means:
Standard Error of the Sample Means:
13. Inference for Two Independent Samples
Technical Detail: Technically, the two sample test
statistic does not follow the t-distribution, but it's
close enough for our purposes. (You don't need to
comment on this.)
14. Inference for Two Independent Samples
Because the computation for the two-sample test
(and confidence interval) is so complicated, we use
the calculator to do the calculations.
Just say no to pooling!
The calculator uses a very complicated formula to find the degrees
of freedom. You don't need to write down this formula!
15. Inference for Two Independent Samples
The book presents a more conservative method to
compute the degrees of freedom. You don't need to
worry about how to do this! Just use the calculator.
16. Inference for Two Independent Samples
We can also use the calculator to compute a two-
sample confidence interval.
Degrees of freedom
Confidence
Don't pool here Interval
either.
17. Inference for Two Independent Samples
Let's try another example.
An educator believes that new reading activities for elementary school
children will improve reading comprehension scores. She randomly assigns
third graders to an eight-week program in which some will use these activities
and others will experience traditional teaching methods. At the end of the
experiment, both groups take a reading comprehension exam. Their scores
are shown in the back-to-back stem-and-leaf display. Do these results
suggest that the new activities are better? Test an appropriate hypothesis
and state your conclusion.
19. Inference for Two Independent Samples
two samples difference in means two means
matched pairs mean difference one mean
20. Inference for Two Independent Samples
American League baseball teams play their games with the designated hitter
rule, meaning that pitchers do not bat. The league believes that replacing the
pitcher, traditionally a weak hitter, with another player in the batting order
produces more runs and generates more interest among fans. Below are the
average numbers of runs scored in American League and National League
stadiums for the first half of the 2001 season.
