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Question # Suppose you are testing whether the lifetime of 4 different types of lightbulbs. You run 5 experiments each. You get a mean square error of MS_{E}=1.7

Comparing two groups
ANSWERED Suppose you are testing whether the lifetime of 4 different types of lightbulbs. You run 5 experiments each. You get a mean square error of $$\displaystyle{M}{S}_{{{E}}}={1.75}$$ and you reject the null hypothesis. The sample means you obtain for each lightbulb are below:
Type 1: 20
Type 2: 30
Type 3: 28
Type 4: 22
Which pairs are significantly different at the $$\displaystyle{5}\%$$ significance level? Select all that apply. 2020-10-21

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.