Question

In an experiment to investigate the performance of four different brands of spark plugs intended for use on a 125-cc motorcycle, five plugs of each brand were tested, and the number of miles (at a constant speed) until failure was observed. A partially completed ANOVA table is given.

Fill in the missing entries, and test the relevant hypotheses using a .05 level of significance. (Give the answer to two decimal places.)

Source of Variation	df	Sum of Squares	Mean Square	F
Treatments	1	2	3	4
Error	5	236682.04	6
Total	7	313520.78

Answer 1

SOLUTION:

The hypothesis in this case would be:

H₀: There is no significant difference between the four different brands of spark plugs

against H₁: not H₀

Five plugs of each of the 4 different brands of spark plugs were tested.

So we have 20 observatons. Thus, n = 20.

• Thus total df will be n-1 = 19
• The treatments are the brands of spark plugs. Thus there are 4 treatments. Hence treatment df = 4-1 = 3
• Error df = Total df - treatment df = 19 - 3 = 16

Total sum of squares = 313520.78 and error sum of squares = 236682.04

Thus, treatment sum of squares = total sum of squares - error sum of squares

• Treatment sum of squares = 313520.78 - 236682.04 = 76838.74

• Mean Squares for treatments = Treatments sum of squares / Treatment df = 76838.74 / 3 = 25612.91
• Mean Squares for error = error sum of squares / error df = 236682.04 / 16 = 14792.63

• F-Ratio will be : Mean Squares for treatments / Mean Squares for error = 25612.91 / 14792.63 = 1.73

Thus the completed table looks like:

Source of Variation	df	Sum of Squares	Mean Square	F - Ratio
Treatments	3	76838.74	25612.91	1.73
Error	16	236682.04	14792.63	-
Total	19	313520.78	-	-

The critical value is F_{3, 16} for 0.05 level of significance is 3.007

Since the calculated F-Ratio is less than the critical vale, we have insufficient evidence to reject H₀ at 5% level of significance,

CONCLUSION:There is no significant difference between the four different brands of spark plugs.