Exploratory Data Analysis
1.3. EDA Techniques
1.3.5. Quantitative Techniques
Test if two population means are equal
The two-sample t-test
Cochran, 1989) is used to determine if two population means
are equal. A common application is to test if a
new process or treatment is superior to a current process
There are several variations on this test.
The two-sample t-test for unpaired data is defined as:
|Two-Sample t-Test Example||
The following two-sample t-test was generated for the
AUTO83B.DAT data set. The data set
contains miles per gallon for U.S. cars (sample 1) and for
Japanese cars (sample 2); the summary statistics for each
sample are shown below.
SAMPLE 1: NUMBER OF OBSERVATIONS = 249 MEAN = 20.14458 STANDARD DEVIATION = 6.41470 STANDARD ERROR OF THE MEAN = 0.40652 SAMPLE 2: NUMBER OF OBSERVATIONS = 79 MEAN = 30.48101 STANDARD DEVIATION = 6.10771 STANDARD ERROR OF THE MEAN = 0.68717
We are testing the hypothesis that the population means are equal for the two samples. We assume that the variances for the two samples are equal.
H0: μ1 = μ2 Ha: μ1 ≠ μ2The absolute value of the test statistic for our example, 12.62059, is greater than the critical value of 1.9673, so we reject the null hypothesis and conclude that the two population means are different at the 0.05 significance level.
In general, there are three possible alternative hypotheses and rejection regions for the one-sample t-test:
For our two-tailed t-test, the critical value is t1-α/2,ν = 1.9673, where α = 0.05 and ν = 326. If we were to perform an upper, one-tailed test, the critical value would be t1-α,ν = 1.6495. The rejection regions for three posssible alternative hypotheses using our example data are shown below.
Two-sample t-tests can be used to answer the following
Confidence Limits for the Mean
Analysis of Variance
|Case Study||Ceramic strength data.|
|Software||Two-sample t-tests are available in just about all general purpose statistical software programs. Both Dataplot code and R code can be used to generate the analyses in this section. These scripts use the AUTO83B.DAT data file.|