Calculating the t Value from Variance Components

This question asks us to find the 't value' given the 'between-groups variance' and 'within-groups variance'. These variance components are commonly associated with statistical tests that compare the means of different groups.

Understanding Between-Groups and Within-Groups Variance

• Between-groups variance: This represents the variation in scores or measurements that is due to the different groups being compared. It reflects the effect of the factor or independent variable being studied.
• Within-groups variance: This represents the variation within each group. It reflects the random error or individual differences that are not explained by the group membership. It's often considered a measure of the variability inherent in the data.

In statistical analysis, particularly in the context of comparing group means, these variances are used to calculate test statistics.

Connecting Variances to Statistical Tests

While the t-test directly compares the means of two groups using their difference and standard error, the 'between-groups variance' and 'within-groups variance' are fundamental components in the calculation of the F-statistic used in Analysis of Variance (ANOVA). The F-statistic is calculated as the ratio of the between-groups variance to the within-groups variance.

In the specific case of comparing only two groups, there is a direct relationship between the F-statistic and the t-statistic: $F = t^2$. Therefore, if we can calculate the F-statistic from the given variances, we can find the t-statistic by taking the square root of the F-statistic.

Step-by-Step Calculation of the t Value

Given:

• Between-groups variance = 200
• Within-groups variance = 50

Step 1: Calculate the F-statistic

The F-statistic is the ratio of the between-groups variance to the within-groups variance:

$$F = \frac{\text{Between-groups variance}}{\text{Within-groups variance}}$$

Substituting the given values:

$$F = \frac{200}{50}$$

$$F = 4$$

Step 2: Calculate the t value from the F-statistic

For a comparison involving two groups, the relationship between the F-statistic and the t-statistic is $F = t^2$. To find the t value, we take the square root of the F value:

$$t = \sqrt{F}$$

Substituting the calculated F value:

$$t = \sqrt{4}$$

$$t = 2$$

Final Answer

Based on the calculation, the 't value' is 2. This corresponds to one of the provided options.

Revision Table: Key Concepts

Additional Information: t-test vs. ANOVA

The t-test is specifically designed to compare the means of two groups. ANOVA (Analysis of Variance) is a more general statistical method used to compare the means of two or more groups. When comparing exactly two groups, the results from an independent samples t-test and a one-way ANOVA are equivalent, with the t-statistic squared equaling the F-statistic. This question uses terms typically associated with ANOVA variance components ('between-groups variance' and 'within-groups variance') to lead to a 't value', leveraging this relationship for the two-group case.