Two Independent Mean Test

μ1 = μ2

1. Whassup? (a.k.a., The Hypotheses)

Before:

H0: μ = 5.27

Ha: μ≠ 5.27 (or , <)

Now:

H0: μ 1– μ 2= 2

Ha: μ 1– μ 2≠ 2 (or > , <)

So for this type of problem, the key is that we will have TWO INDEPENDENT POPULATIONS that we are sampling from. We will have TWO SAMPLES.

Actually, most of the time the number on the end will be 0.

So we will mostly be doing:

H0: μ 1 = μ 2
Ha: μ 1≠ μ 2 (or < or >)

Example: The average heights of men vs. the average heights of women, or the average weight of bottles produced at two competing factories, or the average age of trees in China vs. the average age of trees in the U.S., etc. etc. etc!

2. The Assumptions

1. Two random samples

2. Two independent samples

3. Both sample sizes > 30 or both populations are approximately normal

Which one is new?

3. The Test Statistic

With the assumptions above met, if we know σ1 and σ2 then we use:

(OR if μ 1= μ 2)

If we don’t know σ1or σ2 then we use:

with df = from the calculator only!

The actual formula isheinous.

Basically, if we meet all the assumptions then thesampling distribution of the ’s has mean , standard deviation,and is normally distributed.

If we don’t know the σ’s then we estimate with s’s and we use t.

4. A Confidence Interval??

Okay…so from now on, we will be introducing a new confidence interval every time we introduce a new test!

Remember, here’s what’s on the sheet:

Confidence interval: statistic ± (critical value) • (standard deviation of statistic)

Confidence interval: () ± (tcrit) •

where the df for our tcrit comes from the calculator.

Same assumptions apply!!

5. Example.

11.6 The paper “Effects of Fast-Food Consumption on Energy Intake and Diet Quality Among Children in a National Household Survey” investigated the effect of fast-food consumption on other dietary variables. For a random sample of 663 teens who reported that they did not eat fast food during a typical day, the mean daily calorie intake was 2258 and the sample standard deviation was 1519. For a separate random sample of 413 teens who reported that the did eat fast food on a typical day, the mean calorie intake was 2637 and the standard deviation was 1138.

a. Do a test to see if there is evidence of a difference in the calorie intake between the two populations of teens.

b. Construct a 95% confidence interval for the difference between the two calorie intakes. You do not need to recheck your assumptions in this case.