Chapter 13: Inferences about Comparing Two PopulationsLecture 8b

Date: 15th November 2015Instructor: Naveen Abedin

Recap

• To conduct hypothesis tests for “difference in two population means”, a t-statistic must be used when population means and variances are all unknown.

• There are two types of t-statistics: i. Equal-variances t-statisticii. Unequal-variances t-statistic• Since population variances are unknown, we have to

conduct a “difference in two population variances” hypothesis test to understand whether the population variances are statistically equal or not. Based on the results, we use the appropriate t-statistic to carry out the “difference of means” hypothesis test.

Example 1’s statistics

Example 1: Confidence Interval

• Confidence interval estimate in a two population test estimates the difference between the two means. Here, since population 1 is returns from direct purchase and population 2 is returns from broker purchase, this confidence interval is trying to estimate the difference in returns.

Example 2’s statistics

Some pointers to note• Both the equal-variances test statistic and the

unequal variances test statistic requires the population to be normally distributed. We can check whether these requirements have been satisfied through visual representation of the data in the form of a histogram. If the histogram assumes a normal (or somewhat normal shape), then it is appropriate to use the t-statistic.

• (look in pg 452-453 of textbook for histograms)

Some pointers to note (cont.)

• As seen before, the test statistic to be used for a “difference in population means” test depends on whether or not the population variances are statistically equal or unequal. It is, however, more advantageous to pool sample data under the assumption that the two samples were drawn from populations with a common variance. Combining the samples increases the accuracy of the estimate. This is especially desirable in situations when one or both of the samples are not large enough.

• The pooled variance estimator is said to be an exact procedure, meaning the underlying calculations are perfectly accurate if the assumptions are true.

• It is consistent with other statistical procedures like ANOVA and regression, meaning is we use pooled variance the conclusions would be exactly the same as any one of these other procedures mentioned.

• Be warned: we can get very misleading results if used in the wrong situation.

Observational vs. Experimental data

• Observational study is a study where researchers simply collect data based on what is seen and heard, and infer based on the data collected. Researchers should not interfere with the subjects or variables in any way.

• In observational study, the variables of interest are simply observed (e.g. if a variable of interest is age, simply ask the people in your sample what their ages are). A researcher, also cannot add any extra information or assumptions. All information must be collected solely through observation.

Observational vs. Experimental data (cont.)

• An experiment is a method of applying treatments to a group and recording the effects. A good group experiment will have two basic elements: a control and a treatment. The control is the group that remains untreated throughout the duration of the experiment (i.e. they are not introduced to any artificial situation, like taking a new drug). The treatment is the group upon whom the experiment is conducted (they are the ones who take the new drug).

Observational vs. Experimental data (cont.)

• An observational study is conducted between two populations:

• Population 1: Consumers of high-fiber breakfast cereal

• Population 2: Non-consumers of high-fiber breakfast cereal.

• Theory to test: The study was generated to see if people who consumed high-fiber breakfast cereal necessarily consume fewer calories at lunch.

Observational vs. Experimental data (cont.)

• An unequal variances test was conducted for the difference in population means (mean calories consumed at lunch):

• p – value = 0.0193 < 0.05, so reject the null hypothesis. Therefore, the test shows that people who consume high-fiber cereal consume fewer calories at lunch and thus do not gain so much weight.

Observational vs. Experimental data (cont.)

• However, the test might be showing such favorable results towards high-fiber breakfast cereal for entirely different reasons that high-fiber dietary intake.

• People who eat fewer calories at lunch are probably more-health conscious than people who consume more calories, and the former kind of people are also more likely to consume high-fiber breakfast cereals as a part of a healthy breakfast.

Observational vs. Experimental data (cont.)

• To overcome this problem, it is better to conduct an experimental study.

• Suppose a 150 people are randomly selected to participate in an experiment where 75 of them are assigned to eat high-fiber cereal for breakfast and the other 75 eats something else. Then the number of calories consumed by each group are recorded. Since people have been randomly assigned to each group, on average both groups will be similar in all other dimensions, including health consciousness. (especially if the samples are large).

Observational vs. Experimental data (cont.)

• Experimental data however are usually more expensive than observational data,.

• In many cases it is also not possible to conduct experimental studies. Suppose you want to determine whether an undergraduate degree in engineering better prepared students for an MBA than an arts degree does. In a controlled experiment, one group of student would be assigned to graduate with an engineer degree and then enroll for MBA, and the other group would be assigned to study arts and then enroll for MBA. Of course, this kind of an experiment is impossible to conduct, as no experimental subjects will be willing to commit 5 to 6 years to an experiment.

Observational vs. Experimental data (cont.)

• Thus, we have no choice but to gather information through observational study to assess whether an engineering degree better prepares students for MBA than arts students. If it observed however that engineering students perform better than arts, the cause may not be the degree, but rather the profile of the student. It may be true that better students tend to choose engineering as their undergraduate major and better students achieve higher grades in all programs, including MBA programs.