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

Answer 1

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.