Question
A researcher wanted to study the effects of different types of cooling fans on the
temperature of the different types of computers. The following table gives the observation on temperature of a computer chip when five different types of cooling fans were tried on each of the six different computers:
| Cooling Fan | Computer |
| ||||
| A | B | C | D | E | F | |
| I II III IV V | 26 26 24 24 25 | 18 22 19 21 19 | 23 28 22 23 24 | 12 21 21 18 15 | 21 28 24 19 22 | 15 22 22 20 18 |
Assuming that the effects of each cooling fan and computer are normally distributed with approximately equal variances, analyse the design at 5% level of significance. Test whether the effect on the temperature due to different types of (i) cooling fans and (ii) computers are significant or not at 5% level of significant. If significant, do the pair-wise comparison between them.
Answer :
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Skip in 10
We are asked to analyze the effect of cooling fans and computers on the temperature using the given data. This is a classic two-way ANOVA without replication problem. Let's solve it manually step by step. --- ### Step 1: Organize the data We have 5 cooling fans (I, II, III, IV, V) and 6 computers (A–F). Let’s arrange the data in a matrix: | Fan \ Computer | A | B | C | D | E | F | | -------------- | -- | -- | -- | -- | -- | -- | | I | 26 | 18 | 23 | 12 | 21 | 15 | | II | 26 | 22 | 28 | 21 | 28 | 22 | | III | 24 | 19 | 22 | 21 | 24 | 22 | | IV | 24 | 21 | 23 | 18 | 19 | 20 | | V | 25 | 19 | 24 | 15 | 22 | 18 | --- ### Step 2: Compute totals and means #### (a) Total for each fan (row totals) $$ \text{Row total } T_i = \sum \text{temperatures for that fan} $$ * Fan I: 26+18+23+12+21+15 = 115 * Fan II: 26+22+28+21+28+22 = 147 * Fan III: 24+19+22+21+24+22 = 132 * Fan IV: 24+21+23+18+19+20 = 125 * Fan V: 25+19+24+15+22+18 = 123 Grand total $G = 115 + 147 + 132 + 125 + 123 = 642$ Number of observations: $n = 5 \text{ fans } \times 6 \text{ computers } = 30$ Grand mean: $$ \bar{X} = \frac{642}{30} = 21.4 $$ --- #### (b) Total for each computer (column totals) * Computer A: 26+26+24+24+25 = 125 * Computer B: 18+22+19+21+19 = 99 * Computer C: 23+28+22+23+24 = 120 * Computer D: 12+21+21+18+15 = 87 * Computer E: 21+28+24+19+22 = 114 * Computer F: 15+22+22+20+18 = 97 --- ### Step 3: Compute total sum of squares (SST) $$ SST = \sum X_{ij}^2 - \frac{(G)^2}{n} $$ #### (a) Compute $\sum
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