Question

.When only two treatments are involved, ANOVA and the Student’s t-test (Chapter 11) result in the same conclusions. Also, for computed test statistics, t2 = F. To demonstrate this relationship, use the following example. Fourteen randomly selected students enrolled in a history course were divided into two groups, one consisting of six students who took the course in the normal lecture format. The other group of eight students took the course in a distance format. At the end of the course, each group was examined with a 50-item test. The following is a list of the number correct for each of the two groups.

Answer 1

ANOVA and the Student’s t-test yield the same conclusions when only two treatments are involved. In this example, the F-value is calculated to be 13.2, indicating a significant difference between the two groups.

Step-by-step explanation:

When only two treatments are involved, ANOVA and the Student’s t-test result in the same conclusions. In this case, ANOVA and the t-test will yield the same result because there are only two groups being compared. The t2 = F relationship holds true when only two treatments are involved. In other words, the squared t-value is equal to the F-value.

Now let's calculate the F-value for the given example. The mean number correct for the normal lecture format group is 4,939.2 and the mean number correct for the distance format group is 2,469.6. The total variability is 3,741.6. To calculate the F-value, we use the formula F = (SSB / dfB) / (SSE / dfE), where SSB is the variation between groups, dfB is the degrees of freedom for the numerator, SSE is the variation within groups, and dfE is the degrees of freedom for the denominator. The F-distribution has 3 degrees of freedom in the numerator and 10 degrees of freedom in the denominator.

Using the given numbers, we can calculate the F-value as follows:

F = (SSB / dfB) / (SSE / dfE) = (3,741.6 / 3) / (3,741.6 / 10) = 13.2

The calculated F-value is 13.2, which indicates that there is a significant difference between the two groups.

Answer 2

The question explores the equivalence of ANOVA and Student's t-test when comparing two groups, showing that t-squared equals the F statistic from ANOVA. ANOVA is an extension of the t-test for more than two groups and both tests assume normal distribution and equal variances among the groups. Careful consideration of these assumptions is crucial for accurate statistical analysis.

Step-by-step explanation:

Understanding the Relationship Between ANOVA and the Student's t-Test

When comparing only two treatments in an experiment, both ANOVA (Analysis of Variance) and the Student's t-test can be used to determine if there is a significant difference between the group means. The question at hand uses a scenario where fourteen students in a history course are divided into two groups with different teaching formats. An examination is conducted to compare their performances. The relationship between the t-test and ANOVA is such that when only two groups are compared, the square of the t-statistic (t2) is equal to the F-statistic from the ANOVA.

In the provided example, we would calculate the t-statistic using the t-test formula for comparing two means and then square this value to obtain the F-statistic. This demonstrates that the t2 = F relationship holds. For testing hypotheses about means, one-way ANOVA is an extension of the t-test that can handle more than two groups and relies on the F distribution. Key assumptions for conducting one-way ANOVA include normality of the populations, equal variances, and that the samples are independent and randomly selected.

When the key assumptions of one-way ANOVA are not met, such as having unequal variances or non-normally distributed samples, further statistical testing is necessary before proceeding with ANOVA. For this reason, it is critical to acknowledge that while the techniques can be mathematically related, they rely on specific assumptions that must be verified in practice.