Step 1

From the provided data we have,

\(\displaystyle{M}{S}_{{{E}}}={1.75}\)

\(\displaystyle{d}{f}_{{{t}}}={4}\ \times\ {5}={20}\)

\(\displaystyle{d}{f}_{{{w}}}={20}\ -\ {4}={16}\)

\(\displaystyle{k}=\) number of groups \(\displaystyle={4}\)

\(\begin{array}{|c|c|} \hline \text{Type 1} & \text{Type 2} & \text{Type 3} & \text{Tepe 4} \\ \hline 20 & 30 & 28 & 22 \\ \hline \end{array}\)

Step 2

Using Tukey’s HSD:

We first compute the Tukey’s HSD as below:

\(\displaystyle{D}={q}_{{\alpha,\ {k},\ {d}{f}_{{{w}}}}}\ \sqrt{{{\frac{{{M}{S}_{{{E}}}}}{{{n}}}}}}\)

\(\displaystyle={4.046}\ \sqrt{{{\frac{{{1.75}}}{{{5}}}}}}\) (from studentized q - table)

\(\displaystyle={2.393646}\)

(Where n represent the number of observations in each group).

The absolute differences in group means is computed as below:

\(\begin{array}{|c|c|} \hline \text{Groups} & \text{|Differences in group means|} \\ \hline 1-2 & 10 \\ \hline 1-3 & 8\\ \hline 1-4 & 2\\ \hline 2-3 & 2 \\ \hline 2-4 & 8\\ \hline 3-4 & 6\\ \hline \end{array}\)

We say that the groups are significantly only when the absolute difference in group mean is larger than the HSD value.

Thus, this difference is larger for Group 1 and 2, Group 1 and 3, Group 2 and 4 and Group 3 and 4

Thus, the pairs that are significantly different at the \(\displaystyle{5}\%\) significance level are,

\(\begin{array}{|c|c|} \hline \text{Type 1-2} & \text{Type 1-3} & \text{Type 2-4} & \text{Type 3-4} \\ \hline \end{array}\)

and other groups are not significant.