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.

Inference for Two Independent Samples

You should use more descriptive subscripts than "1" and "2" when doing an actual problem.

Pop Pop Sample SamplePopulation Variable Mean S.D. Mean S.D.

1 X1 µ1 σ1 x1 s1 2 X2 µ2 σ2 x2 s2

Conditions:

Inference for Two Independent Samples

σ from at least one population is unknown.4)

Sample size is less than 1/10 populationsize.

3)

Both sampling distributions are normallydistributed.

2)

SRS from both populations1)

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:

Inference for Two Independent Samples

H0: µ1 – µ2 = 0H0: µ1 = µ2 or

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.

Inference for Two Independent Samples

Here is the data: Brand Name Generic

194.0 190.7205.5 203.5199.2 203.5172.4 206.5184.0 222.5169.5 209.4

What should we do first?

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.)

Inference for Two Independent Samples

Why is this situation independent samples and not matched pairs?

Inference for Two Independent Samples

Perform the hypothesis test, using the inference toolbox.

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:

Inference for Two Independent Samples

So, our test statistic will have the form:

Inference for Two Independent Samples

Terminology Note: Standard Deviation of the Sample Means:

Standard Error of the Sample Means:

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.)

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!

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.

Inference for Two Independent Samples

We can also use the calculator to compute a two-sample confidence interval.

Don't pool here either.

Confidence Interval

Degrees of freedom

Inference for Two Independent SamplesInference 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.

Inference for Two Independent Samples

Inference for Two Independent Samples

two samples difference in means two means

matched pairs mean difference one mean

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.

Inference for Two Independent Samples

Inference for Two Independent Samples