RES/342 Week Two
TWO OR MORE SAMPLE HYPOTHESIS TESTING
Last week, we reviewed the 5-step procedure for performing a simple hypothesis test of a single mean. In this week, we examine hypothesis testing under several other conditions including large and small samples, comparing two independent samples, two dependent samples, Analysis of Variance (ANOVA), and proportions.
This Week in Relation to the Course
"Hypothesis testing refers to a general class of procedures for weighing the strength of statistical evidence-more specifically, for determining whether the evidence supporting one hypothesis over the other is sufficiently strong" (Glasserman, 2001, ¶1). While the 5-step procedure for performing various types of hypothesis tests is the same, the selection of the test statistic (step 3) depends on the nature of the hypothesis and the test data. Here we study how various test conditions alter the determination of the test statistic.
Hypothesis Testing Conditions
Two large, independent samples
It is common to sample from two different populations to determine if the populations have the same mean. In this condition, when the sample size is sufficiently large (30 or more), and the samples are independent. The test statistic is the z statistic, calculated as:
Because the sample size is large enough, this formula works effectively with the standard deviation of the sample. This is fortunate because we usually do not know the standard deviation of the population from which the sample was drawn The formula uses the difference between the two sample means divided by the square root of the variance of the distribution of differences in sample means. Note that if the means are indeed equal, then the difference between the two sample means will be zero.
When the sample size is less than 30, and we are relying on the sample standard deviation, the test statistic is the t statistic and is calculated as follows:
The assumptions that accompany this scenario are a) the populations are normally distributed, b) the populations are independent and, c) the standard deviations of the sampled populations are equal. Given that we are using a sample to draw conclusions about a population, it is entirely likely that there is little knowledge of the nature of the population. For that reason, to assure equal standard deviations, s2p, a pooled value, is used. The pooled value comes from a formula that in effect provides an "average" value for the variance of each population:
Two sample test of proportions
Another commonly used test uses sample proportions to determine if the populations from which they are drawn are different.
The formula is:
This test also requires a "pooled" proportion, pc. The pooling is accomplished with:
When the populations from which we are sampling are dependent, a paired sample is used to generate the value of the test statistic, t:
The numerator is the average of the difference of the paired samples; the denominator the standard deviation of the differences of the paired samples divided by the square root of the number of pairs.
An example of the paired t test would be to compare the cost of a renting a car from Hertz compared with Avis in the same 10 cities. The cost would depend on which company you rented from.
Analysis of Variance (ANOVA) is used to test the equality of the means of three or more groups based on sample data. ANOVA is really a test of means even though it sounds like a test of variance. Building on the hypothesis testing techniques discussed on the first two weeks, ANOVA uses an F test (rather than a Z or t test) to compare multiple means. A hypothesis test using an F statistic can determine whether three or more sample means are statistically significantly different from each other, otherwise one may conclude that they are from the same population. The concepts of variance and sums of squared differences used in ANOVA can be used to evaluate the output of linear regression, the topic of Week Four.
Practical Application and Questions for Thought
The most critical aspect of hypothesis testing is choosing the correct measure of the test statistic. In all other respects, the 5-step hypothesis testing protocol is exactly the same. Given a business scenario, the following questions need to be asked and answered:
Are the data independent or dependent? Dependent data suggests the paired t test is appropriate. Is the problem a test of means or proportions? Is the sample small (<30) or large?
Consider the following example. A baseball team wants to test the effect of using an iridescent baseball instead of a plain white one. Would batting averages be higher if the league changed to an iridescent baseball? The starting lineup goes into a batting cage (one at a time) to face a pitching machine set to release a straight fastball at 88 mph. Each batter gets 10 swings at a standard baseball. The process is repeated with the iridescent ball.
Since the starting line up is nine players, the sample size is considered small. Since the outcomes depend on which ball is being hit, the samples are dependent. Therefore, the appropriate test in this case is a paired t test. By analyzing the data in any scenario, the appropriate test statistic should reveal itself.
When comparing the salaries of a sample of male executives and a sample of female executives, are the data independent or dependent?
How Readings Solidify Concepts
This week's readings show many variations of hypothesis testing. Most are built on an understanding of descriptive statistics studied in previous weeks. Most of the examples are accompanied by a small diagram depicting a normal distribution and markings for the critical value and observed value. Drawing a diagram is a great aid to visualizing what the research question really asks. The simulation provides an example of how different types of data require different test statistics, yet the hypothesis testing procedure is the same.
Performing tests of hypotheses, determining critical values, calculating the test statistic, these things are not overly difficult and better still, can be solved with Excel® or statistical software packages. However, the researcher must first know what kind of test is required, and this is determined by the data being studied and the question being asked.
Glasserman, P. (2001). Hypothesis testing. Retrieved October 9, 2004, from http://www-1.gsb.columbia.edu/faculty/pglasserman/B6014/HypothesisTesting.pdf
The expert examines hypothesis testing for ANOVA. Neat, step-by-step explanation is provided.