CHAPTER 14: t TEST FOR TWO INDEPENDENT SAMPLES

Key Terms

Two independent samples - Observations in one sample are not paired, on a one-to-one basis, with observations in the other sample.

______

Sampling distribution of X1-X2 -- Difference between sample means based on all possible pairs of random samples---of given sizes---from two underlying populations.

Effect - Any difference between two population means

______

Standard error of X1-X2 - A rough measure of the average amount by which any difference between

sample means deviates from the difference between population means

2

Pooled variance estimate (s p) - The most accurate estimate of the population variance (assumed to be the same for both populations) based on a combination of two sample variances.

Confidence intervals for 1- 2 - A range of values that, in the long run, includes the unknown difference between population means a certain percent of the time.

p-value - The degree of rarity of a test result, given that the null hypothesis is true.

Statistical significance - Not an indication of importance; but merely that the null hypothesis is probably true.

Effect - difference between population means.

Squared point biserial correlation - The population of variance in the dependent variable that can be explained by the independent variable.

Text Review

Section I

Two independent samples occur when (1)_________ in one sample are not paired with observations in the other sample. When a t test is conducted for two independent samples, the difference between population means reflects the (2)________ of the variable being studied. In the example in the text, the variable is (3)____________________. When there is little difference between the two population means, there is little effect.

The null hypothesis states there is (4)______ difference between population means. There are three possible alternative hypotheses. One states that the difference between population means does not equal zero. This would represent a (5)___________________ test. A second possible hypothesis states that the difference is less than zero. This is a one-tailed test with the (6)_________tail critical. The third possibility is a one-tailed test with (7)__________ tail critical which states that the difference between population means (8)_______ zero. When there is concern only about difference in a particular direction, a (9)__________ or one-tailed hypothesis test should be used.

Just as the sampling distribution of the mean (presented in Chapter 13) is not actually constructed, the sampling distribution for the difference between sample means is not constructed. Instead, we rely on statistical theory to provide information about the mean and standard error for the

______

sampling distribution of X1-X2. In practice, there is only one observed difference and the (10)___ is conducted to determine whether it qualifies as a common or rare outcome. In the one-sample case, the mean of the sampling distribution equals the mean of (11)_________. In the two-sample case, the mean of the sampling distribution equals the (12)_______________ between population means.

The sampling distribution of X1-X2 has a standard deviation referred to as the (13)___________ of the difference between sample means. The standard error is a rough measure of the average amount by which any difference between sample means deviates from the difference between (14)_________

____________. The size of the standard error decreases as the sample size (15)________________.

Before the t ratio can be calculated, the standard error must be estimated. The t test assumes that the two populations are (16)__________________.

A confidence interval may be constructed for the difference between population means. The confidence interval is a range of values that, in the long run, includes the unknown difference between population means a certain percent of the time. If both positive and negative values appear in a confidence interval, no single interpretation is possible. The inclusion of a zero value in the range indicates that the variable being studied may have (17)_____________________.

The t test for two independent samples assumes that both underlying populations are (18)_____

_____________________ and have (19)_____________ variances. If sample sizes are (20) _________

___________ and ________, violations of these assumptions will be of little concern.

The pooled variance estimate can be obtained by combining the variance common to both populations. The pooled variance estimate is the most accurate estimate of the variance (assumed to be the same for both populations) based on a combination of the two sample variances. The degrees of freedom for the pooled variance estimate equal the sum of the two sample sizes minus two. Two degrees of freedom are lost because the (21)___ in each of the two samples are expressed as (22)___ from their respective sample means.

Section II

If the researcher does not retain or reject the null hypothesis, but views it with suspicion, depending on the degree of rarity of the test, then (1)_______________________ are being used. Smaller p-values tend to discredit the null hypothesis and support the research hypothesis. Using p-values is a less structured approach to hypothesis testing.

If level of significance is set at .05, any p-value less than .05 implies that the null hypothesis would have been (2)______________________ and any p-value greater than .05 implies that the null hypothesis would be (3)_________________.

Computer generated p-value are more precise, but are interpreted in the same way as those read from the tables in the appendix of the test.

Using the less structured approach of (4)___-________________ allows the researcher to postpone a decision until subsequent investigation can provide further evidence. This is an an attractive option when test results are (5)_______________________.

One disadvantage of this approach is that when a firm decision is not being made to retain or reject the null hypothesis, it is difficult to deal with the concept of type I and (6) _____________ ___ errors.

Results of hypotheses are often described as having statistical significance if the (7)____________ hypothesis has been rejected. Statistical significance indicate that the null hypothesis is probably false, but doesn't indicate whether it is seriously false or mildly false. Statistical significance may lack practical importance because sample size is (8)__________________________.

One way to judge practical importance when large sample sizes are present is to use the squared (9)____________________________________ correlation. The squared point biserial correlation indicates the proportion of variance in the dependent variable that is predictable from the (10)_________________________ variable. Cohen suggested a rule of thumb for interpreting the values of squared point correlation in relation to effect size. However, it is best not to apply any rule of thumb without considering special circumstances that could make even a very small effect very