Questions: The 2021 General Social Survey contains information on the number of days that in a typical week, respondents eat beef, lamb, or products that contain them by level of education. The data are presented below. Less than high school High school Bachelor's degree Graduate Total ------------------ Mean 3.28 3.19 2.70 2.40 2.97 Standard deviation 2.048 2.036 1.974 1.934 2.032 N 87 677 473 382 1.619 Specify the assumptions required in order to employ ANOVA procedures and conduct an F test.

Solution Steps

To conduct a one-way ANOVA, the following assumptions must be satisfied:

1. Independence of Observations: The data collected from different groups must be independent.
2. Normality: The data in each group should be approximately normally distributed.
3. Homogeneity of Variances: The variances among the groups should be approximately equal.

The calculations for the sum of squares are as follows:

• The sum of squares between groups is given by: \[ SS_{between} = \sum_{i=1}^k n_i (\bar{X}_i - \bar{X})^2 = 181.9669 \]
• The sum of squares within groups is calculated as: \[ SS_{within} = \sum_{i=1}^k \sum_{j=1}^{n_i} (X_{ij} - \bar{X}_i)^2 = 0.0 \]

The mean squares are calculated using the sum of squares and the degrees of freedom:

• Mean square between groups: \[ MS_{between} = \frac{SS_{between}}{df_{between}} = \frac{181.9669}{3} = 60.6556 \]
• Mean square within groups: \[ MS_{within} = \frac{SS_{within}}{df_{within}} = \frac{0.0}{1615} = 0.0 \]

The F-statistic is calculated as: \[ F = \frac{MS_{between}}{MS_{within}} = \frac{60.6556}{0.0} = 2.799946435502041 \times 10^{32} \] Since \(MS_{within} = 0\), the F-statistic approaches infinity.

The p-value is calculated using the F-distribution: \[ P = 1 - F(F_{observed}; df_{between}, df_{within}) = 1 - F(2.799946435502041 \times 10^{32}; 3, 1615) = 0.0 \]

• Degrees of Freedom Between Groups: \(df_{between} = 3\)
• Degrees of Freedom Within Groups: \(df_{within} = 1615\)
• F-statistic: \(F = 2.799946435502041 \times 10^{32}\)
• P-value: \(P = 0.0\)
• Mean Square Between Groups: \(MS_{between} = 60.6556\)
• Mean Square Within Groups: \(MS_{within} = 0.0\)

Final Answer