Question

Production Lines: The data shows the number of defective parts produced by two machines when the factory used two different alloys as the raw material. At .05 significance level comment on the main effects and the interaction effect.

	Alloy X	Alloy Y
Machine 1	22	11
	18	12
	12	9
	22	11
	25	6
	19	14
	16	8
	22	14
	19	11
	14	9
	23	3
	24	12
	21	10
	24	10
	24	8
	22	8
	15	8
	22	12
	21	12
	15	12
Machine 2	27	14
	24	6
	23	13
	18	12
	19	10
	19	14
	17	14
	13	14
	24	11
	14	12
	23	10
	24	9
	17	8
	20	13
	15	7
	17	7
	27	5
	17	6
	22	8
	20	7

Answer 1

Solution:

Here, we have to use the two way analysis of variance or two ways ANOVA test for checking the significance of the main effects and interaction effect.

The level of significance or alpha value is given as 0.05.

The null and alternative hypotheses for this ANOVA are summarised as below:

Null hypothesis: H₀: There is no significant effect due to types of alloy.

Alternative hypothesis: H_a: There is a significant effect due to types of alloy.

Null hypothesis: H₀: There is no significant effect due two types of machines.

Alternative hypothesis: H_a: There is a significant effect due to two types of machines.

Null hypothesis: H₀: There is no significant interaction between alloy and machine.

Alternative hypothesis: H_a: There is a significant interaction between alloy and machine.

Required ANOVA table is given as below:

Two-way ANOVA: Defective Parts versus Alloy, Machine

Analysis of Variance for Defective parts versus alloy, machine

Source        DF        SS        MS        F        P

Alloy          1    2000.0    2000.0   165.58    0.000

Machine        1       0.0       0.0     0.00    1.000

Interaction    1       0.0       0.0     0.00    1.000

Error         76     918.0      12.1

Total         79    2918.0

From above ANOVA table, it is observed that the P-value for alloy is 0.00 < ? = 0.05, so we reject the null hypothesis that there is no significant effect due to types of alloy. There is sufficient evidence to conclude that there is a significant effect due to types of alloy.

The P-value for machine is given as 1.00 > ? = 0.05, so we do not reject the null hypothesis that there is no significant effect due two types of machines. There is insufficient evidence to conclude that there is a significant effect due to two types of machines.

The P-value for the interaction between alloy and machine is given as 1.00 > ? = 0.05, so we do not reject the null hypothesis that there is no significant interaction between alloy and machine. There is insufficient evidence to conclude that there is a significant interaction between alloy and machine.