1. Comparing Two Populations or Treatments with Significance Testing
10.2B
Comparing Two Populations or Treatments with Significance Testing
AP $20 due 29th!

2. Pooling
Guess what? You don’t use it! We have technology that uses the better procedure. Pooling assumes that the two populations have the same variance. This procedure pools the two sample variances to estimate the common population variance. n1+n2-2 degrees of freedom if the two population variance really are equal and the population distributions are exactly Normal.

3. Objectives
Write your hypotheses
Check conditions
Perform mechanics
Make a decision linked in context

4. Inference about the difference between TWO means
You are concerned about one of two things:
Comparing two populations
Establishing causation
How the experiment was designed informs what you are able to conclude!

5. Hypothesis Test Comparing Two Means Example—Root Beer
We would like to compare the mean fill of 16 ounce bottles of root beer from two adjacent filling machines. As a plant owner I want these machines to be the same. But I am worried that they are actually different.

6. So, I take a sample from each population!
A sample of 35 cans from machine A gave a mean of oz with a stdev oz and a sample of 31 cans from machine B gave a mean of oz with stdev oz . State, perform and interpret an appropriate hypothesis test using the 0.05 level of significance.

7. If we meet the RIN conditions for both sets of data, then we can run a two-sample t-test to compare means.
with degrees of freedom found by technology or found by table using the smaller of 𝑛 1 −1 or 𝑛 2 −1

8. Make sure you identify all variables!
A sample of 35 cans from machine #1 gave a mean of oz with a stdev oz and a sample of 31 cans from machine #2 gave a mean of oz with stdev oz . State, perform and interpret an appropriate hypothesis test using the 0.05 level of significance.

9. Hypothesis Test – Cold Medicine
In an attempt to determine if two competing brands of cold medicine contain, on the average, the same amount of acetaminophen, twelve different tablets from each of the two competing brands were randomly selected and tested for the amount of acetaminophen each contains. The results (in milligrams) follow. Use a significance level of 0.01.
Brand A Brand B
517, 495, 503, , 508, 513, 521
503, 493, 505, , 533, 500, 515
498, 481, 499, , 498, 515, 515
State and perform an appropriate hypothesis test.

10. Hypothesis Test – Cold Medicine
Brand A Brand B
517, 495, 503, , 508, 513, 521
503, 493, 505, , 533, 500, 515
498, 481, 499, , 498, 515, 515
State and perform an appropriate hypothesis test.

11. Hypothesis Test – Example 2
The samples were selected independently and randomly. Since the samples are not large, we need to be able to assume that the populations (of amounts of acetaminophen) are both normally distributed.
You MIGHT be given graphs. If so, you MUST refer to those graphs in the check of normality….

12. You see this graph….
You say, “looks linear”
In a NPP, linear is an indication that it’s feasible the data came from a normal population.

13. You have to say “looks linear” for BOTH sets of data
And it’s A-OK to just write on the graph.

14. Maybe you get boxplots….
No outliers, that’s a good sign.
Say so—write on the graphs

15. Maybe all you get is data
Brand A Brand B
517, 495, 503, , 508, 513, 521
503, 493, 505, , 533, 500, 515
498, 481, 499, , 498, 515, 515
Make your own graphs. But do the quick and easy kind.
Brand B 49
3 8 50
8 0 51 52 1 53
3 6 54
Brand A 48 1 49 50
3 3 5 51 7
And don’t forget to say something about them!

16. © 2008 Brooks/Cole, a division of Thomson Learning, Inc.
Suppose you are designing an experiment
to determine if students perform better on tests
when there are no distractions, such as a teacher talking on the phone. You have access to two classrooms and 30 volunteers who are willing to participate in your experiment
Design an experiment so that a two-sample t-test would be the appropriate inference method
Design an experiment so that a paired t-test would be the appropriate inference method.
Which experimental design is better? Explain.
© 2008 Brooks/Cole, a division of Thomson Learning, Inc.

17. © 2008 Brooks/Cole, a division of Thomson Learning, Inc.
AP $20 due 29th!
Measure your pulse rate (beats per minute)
Drink your soda
After 10 minutes, we will measure your pulse rate again
© 2008 Brooks/Cole, a division of Thomson Learning, Inc.

18. © 2008 Brooks/Cole, a division of Thomson Learning, Inc.
What must be true in order to perform a t-test for the
difference between two means?
difference between two proportions?
© 2008 Brooks/Cole, a division of Thomson Learning, Inc.

19. © 2008 Brooks/Cole, a division of Thomson Learning, Inc.
How is the power of the test affected by….
changing sample size?
changing α?
changing β?
© 2008 Brooks/Cole, a division of Thomson Learning, Inc.

20. © 2008 Brooks/Cole, a division of Thomson Learning, Inc.
When you do not have a calculator,
how to do calculate degrees of freedom
for two samples?
© 2008 Brooks/Cole, a division of Thomson Learning, Inc.

21. © 2008 Brooks/Cole, a division of Thomson Learning, Inc.
For each scenario, determine the proper inference procedure
from Chapter 8,9, or 10 you would use. Be specific.
© 2008 Brooks/Cole, a division of Thomson Learning, Inc.

22. © 2008 Brooks/Cole, a division of Thomson Learning, Inc.
Do people smoke less when cigarettes cost more? A random sample of 500 smokers was selected. The number of cigarettes each person smoked per day was recorded over a one-month period before a 30% cigarette tax was imposed and again for one month after the tax was imposed.
Paired t test for the mean difference
© 2008 Brooks/Cole, a division of Thomson Learning, Inc.

23. © 2008 Brooks/Cole, a division of Thomson Learning, Inc.
How much greater is the percent of senior citizens who attend a play at least once per year than the percent of people in their twenties who do so? Random samples of 100 senior citizens and 100 people in their twenties were surveyed.
Two-sample z interval for the difference in proportions
© 2008 Brooks/Cole, a division of Thomson Learning, Inc.

24. © 2008 Brooks/Cole, a division of Thomson Learning, Inc.
You have data on rainwater collected at 16 locations in the Adirondack Mountains of New York State. One measurement is the acidity of the water, measured by pH on a scale of 0 to 14 (the pH of distilled water is 7.0). Estimate the average acidity of rainwater in the Adirondacks.
One sample t interval for the mean
© 2008 Brooks/Cole, a division of Thomson Learning, Inc.

25. © 2008 Brooks/Cole, a division of Thomson Learning, Inc.
Consumers Union wants to see which of two brands of calculator is easier to use. They recruit 100 volunteers and randomly assign them to two equal-sized groups. The people in one group use Calculator A and those in the other group use Calculator B. Researchers recorded the time required for each volunteer to carry out the same series of routine calculations(such as figuring discounts and sales tax, totaling a bill) on the assigned calculator.
Two-sample t interval for the difference between two means
© 2008 Brooks/Cole, a division of Thomson Learning, Inc.