Problem 8 - An Introduction to Psychological Statistics

Question

Administrators at a university want to know if students in different majors are more or less extroverted than others. They provide you with data they have for English majors \((\overline{\mathrm{X}}=3.78, n=45)\), History majors \((\overline{\mathrm{X}}=2.23, n=40)\), Psychology majors \((\overline{\mathrm{X}}=4.41, n=51)\), and Math majors \((\overline{\mathrm{X}}=1.15, n=28)\). You find the \(S S B=75.80\) and \(S S W=47.40\) and test at \(\alpha=0.05\)

Step-by-Step Solution

Answer

Reject the null hypothesis; there's a significant difference in extroversion among majors.

Step 1: Identify the Null and Alternative Hypotheses

The null hypothesis \((H_0)\) states that there are no differences in extroversion levels across the different majors: \(\mu_1 = \mu_2 = \mu_3 = \mu_4\). The alternative hypothesis \((H_1)\) suggests that at least one pair of means is significantly different.

Step 2: Determine the Degrees of Freedom

For the ANOVA test, we calculate the between-groups degrees of freedom \((df_{between})\) as the number of groups minus one: \(df_{between} = k - 1 = 4 - 1 = 3\). The within-groups degrees of freedom \((df_{within})\) is the total number of observations minus the number of groups: \(df_{within} = N - k = (45 + 40 + 51 + 28) - 4 = 160.\)

Step 3: Calculate the Mean Squares

Calculate the Mean Square Between \((MSB)\) by dividing the sum of squares between by the degrees of freedom between: \(MSB = \frac{SSB}{df_{between}} = \frac{75.80}{3} = 25.27.\) Calculate the Mean Square Within \((MSW)\) by dividing the sum of squares within by the degrees of freedom within: \(MSW = \frac{SSW}{df_{within}} = \frac{47.40}{160} = 0.29625.\)

Step 4: Compute the F-statistic

The F-statistic is calculated by dividing the Mean Square Between by the Mean Square Within: \(F = \frac{MSB}{MSW} = \frac{25.27}{0.29625} \approx 85.28.\)

Step 5: Determine the Critical Value

Using the F-distribution table, find the critical F-value at \(\alpha = 0.05\) with \(df_{between} = 3\) and \(df_{within} = 160\). The critical value is approximately 2.66.

Step 6: Make a Decision

Since the calculated F-statistic \(85.28\) is greater than the critical F-value \(2.66\), we reject the null hypothesis. This means there are significant differences in extroversion levels among the majors.

Key Concepts

Degrees of Freedom

In the context of ANOVA (Analysis of Variance) tests in psychology, degrees of freedom help us understand how much information we have available to estimate a parameter. They represent the number of independent values that can vary in an analysis without breaking any given rules or restrictions.
For instance, with our university exercise:

• **Between-groups degrees of freedom (\(df_{between}\))** is defined by the number of groups minus one. With 4 major groups (English, History, Psychology, Math), we have \(df_{between} = 4 - 1 = 3\). This tells us how flexible our data is across different groups.


• **Within-groups degrees of freedom (\(df_{within}\))** is determined by the total number of observations minus the number of groups. With a total of 164 observations, we have \(df_{within} = 164 - 4 = 160\). This shows the variability within each group.

These degrees of freedom are key for calculating values like mean squares and ultimately, the F-statistic.

Null and Alternative Hypothesis

In hypothesis testing within psychology, especially ANOVA, establishing a null and alternative hypothesis is a foundational step. The **null hypothesis (\(H_0\))** asserts that there is no effect or difference; in our case, it claims that students' extroversion levels are the same across all majors. This is symbolically represented as \(\mu_1 = \mu_2 = \mu_3 = \mu_4\).

The **alternative hypothesis (\(H_1\))**, on the other hand, suggests that there is a difference. It proposes that at least one group's extroversion level is significantly different from the others. However, it does not specify which or how many groups differ, leaving that detail up for further examination.
These hypotheses are pivotal because they guide the direction of research and analysis in confirming whether an observed effect in data is likely due to a random chance or a true variation between groups.

F-Statistic

The F-statistic is a key element in ANOVA that helps us understand if the differences observed between group means are statistically significant. It tests the ratio of variance between groups to variance within groups. In our task, the F-statistic allows us to compare the variability in extroversion between majors against the variability within each major.

To calculate the F-statistic, we use the formula: \(F = \frac{MSB}{MSW}\). Where **\(MSB\)** is the mean square between groups, calculated from the sum of squares between divided by its degrees of freedom. And **\(MSW\)** is the mean square within groups, calculated from the sum of squares within divided by its degrees of freedom.
In our example, we found \(F = \frac{25.27}{0.29625} \approx 85.28\). This is a large F-value, indicating greater variance between groups than within, implying significant differences among the extroversion levels of students across different majors.

Critical Value

A critical value in ANOVA refers to the boundary at which we decide whether to reject or fail to reject the null hypothesis. This value is determined by the chosen significance level (\(\alpha\)), typically set as 0.05 for psychological studies, and the degrees of freedom from our analysis.
We obtain the critical value from an F-distribution table using \(df_{between}\) and \(df_{within}\). In our university exercise, these values are 3 and 160, respectively. With an \(\alpha\) of 0.05, the critical value is approximately 2.66.
We compare this critical value with our calculated F-statistic, in this case, 85.28. Since 85.28 is much larger than 2.66, we reject the null hypothesis. This means that the variation in extroversion levels between majors is unlikely due to random chance, pointing to genuine differences in student extroversion by major.