The t-test is used for comparing two means. When there are more than two means analysis of variance is used. We will discuss this statistical procedure later.

The independent variable is nominal and has two categories. The dependent variable is either ratio or interval.

For example: College graduates have higher incomes than non-college graduates. In this hypothesis, college graduate is the independent variable and is nominal with two categories. Income is the dependent variable its level of measurement is ratio.

For example: People who live in dorms have a higher GPA than people who live in private apartments. The independent variable is where you live (either dorm or apartment, which is nominal), and the dependent variable is GPA which is ratio data.

There are three types of t-test.

1. Independent samples: This test procedure compares means for two groups of cases. Ideally, for this test, the subjects should be randomly assigned to two groups, so that any difference in response is due to the treatment (or lack of treatment) and not to other factors. This is not the case if you compare average income for males and females. A person is not randomly assigned to be a male or female. In such situations, you should ensure that differences in other factors are not masking or enhancing a significant difference in means. Differences in average income may be influenced by factors such as education and not by sex alone.

Example: Patients with high blood pressure are randomly assigned to a placebo group and a treatment group. The placebo subjects receive an inactive pill and the treatment subjects receive a new drug that is expected to lower blood pressure. After treating the subjects for two months, the two-sample t test is used to compare the average blood pressures for the placebo group and the treatment group. Each patient is measured once and belongs to one group.

Statistics: For each variable: sample size, mean, standard deviation, and standard error of the mean. For the difference in means: mean, standard error, and confidence interval (you can specify the confidence level). Tests: Levene's test for equality of variances, and both pooled- and separate-variances t tests for equality of means.

Interpretation: Look at the Levene's test to determine which t you will use, then note the significance level of that t and decide whether it must be divided for a one-tailed hypothesis. If the t has significance (i.e. if the significance level is less than 0.05), then write the difference in means (giving the values for each), and note whether the hypothesis is supported or rejected. If the t is not significant, write the difference in means (giving the values for each and noting the difference is not significant) and indicate whether the hypothesis is supported or rejected.

2. One sample t-test: This procedure tests whether the mean of a single variable differs from the population.

Examples: A researcher might want to test whether the average IQ score for a group of students differs from 100. Or, a cereal manufacturer can take a sample of boxes from the production line and check whether the mean weight of the samples differs from 1.3 pounds at the 95% confidence level. Another example would be if we wanted to compare the GPA of a sample to the GPA of the entire student population.

Statistics: For each test variable: mean, standard deviation, and standard error of the mean. The average difference between each data value and the hypothesized test value, a t test that tests that this difference is 0, and a confidence interval for this difference (you can specify the confidence level).

Interpretation: If the t has significance (i.e. if the significance level is less than 0.05), then indicate there are significant differences between the sample mean and the standard (giving the values for each), and note whether the hypothesis is supported or rejected. If the t is not significant, write the difference between the sample mean and the standard (giving the values for each and noting the difference is not significant) and indicate whether the hypothesis is supported or rejected.

3. Paired sample t-test: This procedure compares the means of two variables for a single group. It computes the differences between values of the two variables for each case and tests whether the average differs from 0.

Example: In a study on high blood pressure, all patients are measured at the beginning of the study, given a treatment, and measured again. Thus, each subject has two measures, often called before and after measures. An alternative design for which this test is used is a matched-pairs or case-control study. Here, each record in the data file contains the response for the patient and also for his or her matched control subject. In a blood pressure study, patients and controls might be matched by age (a 75-year-old patient with a 75-year-old control group member).

Statistics: For each variable: mean, sample size, standard deviation, and standard error of the mean. For each pair of variables: correlation, average difference in means, t test, and confidence interval for mean difference (you can specify the confidence level). Standard deviation and standard error of the mean difference.

Interpretation: If the t has significance (i.e. if the significance level is less than 0.05), then indicate there are significant differences between the two means (giving the values for each), and note whether the hypothesis is supported or rejected. If the t is not significant, write the difference between the two means (giving the values for each and noting the difference is not significant) and indicate whether the hypothesis is supported or rejected.

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