Samples were drawn from three populations. The sample sizes were $n _ { 1 } = 6 , n _ { 2 } = 8$, and $n _ { 3 } = 5$. The sample means were $\bar { x } _ { 1 } = 2.79 , \bar { x } _ { 2 } = 1.65$, and $\bar { x } _ { 3 } = 2.13$. The sample standard deviations were $s _ { 1 } = 0.26 , s _ { 2 } = 0.42$, and $s _ { 3 } = 0.49$. The grand mean is $\bar { x } = 2.136316$. Can you conclude that two or more of the population means are different? Use the α = 0.05 level of significance.

Question

Quizplus · Accepted Answer

We will use the one-way ANOVA test to determine if there is a significant difference between the population means. The null hypothesis is that all population means are equal, and the alternative hypothesis is that at least two population means are different.

The test statistic for ANOVA is calculated as:

$$F = \frac{MS_{between}}{MS_{within}}$$

where

$$MS_{between} = \frac{\sum\limits_{i=1}^k n_i(\bar{x_i} - \bar{x})^2}{k-1} $$

and

$$MS_{within} = \frac{\sum\limits_{i=1}^k \sum\limits_{j=1}^{n_i} (x_{ij} - \bar{x_i})^2}{N-k}$$

In this case, we have three samples, so k=3. Using the given values:

$$\bar{x} = 2.136316$$

$$\bar{x_1} = 2.79$$

$$\bar{x_2} = 1.65$$

$$\bar{x_3} = 2.13$$

$$n_1 = 6$$

$$n_2 = 8$$

$$n_3 = 5$$

$$s_1 = 0.26$$

$$s_2 = 0.42$$

$$s_3 = 0.49$$

Using these values, we get:

$$MS_{between} = 1.08937$$

$$MS_{within} = 0.04547$$

Therefore,

$$F = \frac{MS_{between}}{MS_{within}} = 23.9647$$

To determine if this is statistically significant, we compare the calculated F-statistic with the critical value from the F-distribution table with degrees of freedom (k-1, N-k) = (2, 18) at the α = 0.05 level of significance.

The critical value is F = 3.49. Since our calculated F-statistic is greater than the critical value, we reject the null hypothesis and conclude that at least two population means are different.

Samples Were Drawn from Three Populations n1=6,n2=8n _ { 1 } = 6 , n _ { 2 } = 8n1​=6,n2​=8

Samples Were Drawn from Three Populations $n _ { 1 } = 6 , n _ { 2 } = 8$