Barttletts test and F test with 3 variables

Question

I linked a picture with the assignment.

I know how to do F-tests with 2 variances and means, but with 3, I am out of luck. I could do it if I had a dataset in R. but not manual in R or by hand, as it is in this assignment.

Answer 1

2) $F$-test, assume $\alpha=0.05$: $$H_0:\mu_1=\mu_2=\mu_3\\ \ \ \ \ \ \ \ H_1: \text{at least two differ}\\ \begin{align}F&=\frac{MST}{MSE}\approx \frac{\color{red}{0.90231}}{\color{blue}{0.13531}}\approx 6.67,\\ MST&=\frac{SST}{k-1}=\frac{\sum n_i(\bar{y}_i-\bar{\bar{y}})^2}{k-1}\approx \\ &\approx\frac{9(0.7467-0.77002)^2+\cdots +9(1.0889-0.77002)^2}{2}\approx \color{red}{0.90231},\\ MSE&=\frac{\sum (n_i-1)s_i^2}{n-k}=\frac{(9-1)\cdot 0.13655+\cdots+(9-1)\cdot 0.17854}{26-3}\approx \color{blue}{0.13531},\\ F_{\alpha/2,k-1,n-k}&=F_{0.025,2,23}=4.35,\\ F&>F_{\alpha/2} \Rightarrow \text{Reject $H_0$}.\end{align}$$

Answer 2

I assume you want to know whether the three sample variances .137, .084, and .179 in the table of your Question are consistent with equal group variances for Groups 1, 2, and 3.

You are correct that you can't do Bartlett's test from the summary statistics you have. The same is true of Levene's test for equal variances (used when data are not normal).

However, Hartley's F-max test can be based on what you have in the table. It takes the ratio $F_{\text{Max}}$ of the largest of $k$ group variances divided by smallest. For your data $k = 3$ and the ratio is $.179/.084 = 2.13.$ A table of 5% critical values for $k = 3$ and sample sizes 8 and 9 shows that rejection of the null hypothesis of equal variances would require the ratio to exceed about 6 or 7 in order to reject. (The table assumes all three sample sizes are equal, so a little 'flexibility' is in order.)

Hartley's test is not used much these days because it is less powerful for normal data than Bartlett's test, and statistical software assumes complete data are at hand.

In your case it would be safe to say there is no significant difference among the three if there is no significant difference between the smallest and largest of three. The F-test for two sample variances is easy to do with the values you have. Minitab output (slightly abridged for relevance) for a one-sided test is shown below. (The P-value for a two-sided test is twice as large.) It seems your sample variances are nowhere near significantly different.

Statistics

Sample  N  StDev  Variance    
First   9  0.423     0.179     
Second  8  0.290     0.084    

Ratio of standard deviations = 1.460
Ratio of variances = 2.131

Test
                       Test
Method  DF1  DF2  Statistic  P-Value
F         8    7       2.13    0.167