Question

A consumer research organization was interested in the influence of type of water on the effectiveness of a detergent. Test batches of washings were run in four randomly chosen machines having a particular type of water - soft, moderate, and hard. All batches had equal numbers of oil-stained rags and after each washing the number of rags stilled stained was determined. The following results were obtained.

                                     Number of Rags with Stains

              Sample

             Observations        Soft      Moderate      Hard

                      1                   0             4              10

                      2                   1             8                5

                      3                   2             3                7

                      4                   1             5               10

Using an alpha level of .01, would you conclude that the type of water influences the effectiveness of the detergent?

Answer 1

For this we will perform One Way ANOVA

The following table is obtained:

	SOft	Moderate	Hard
	0	4	10
	1	8	5
	2	3	7
	1	5	10
Sum =	4	20	32
Average =	1	5	8
∑​Xij2​=	6	114	274
St. Dev. =	0.816	2.16	2.449
SS =	2	14	18
n =	4	4	4

The total sample size is N=12. Therefore, the total degrees of freedom are:

df_{total} = 12 - 1 = 11

Also, the between-groups degrees of freedom are df_{between} = 3 - 1 = 2, and the within-groups degrees of freedom are:

df_{within} = df_{total} - df_{between} = 11 - 2 = 9

First, we need to compute the total sum of values and the grand mean. The following is obtained

∑​Xij​=4+20+32=56

Also, the sum of squared values is

∑​Xij2​=6+114+274=394

Based on the above calculations, the total sum of squares is computed as follows:

The within sum of squares is computed as shown in the calculation below:

Now that sum of squares are computed, we can proceed with computing the mean sum of squares:

Finally, with having already calculated the mean sum of squares, the F-statistic is computed as follows:

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ₁ = μ₂ = μ₃

Ha: Not all means are equal

The above hypotheses will be tested using an F-ratio for a One-Way ANOVA.

(2) Rejection Region

Based on the information provided, the significance level is \alpha = 0.01α=0.01, and the degrees of freedom are df_1 = 2 and df_2 = 9, therefore, the rejection region for this F-test is R={F:F>Fc​=8.022}.

(3) Test Statistics

(4) Decision about the null hypothesis

Since it is observed that F=13.059>Fc​=8.022, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0022, and since p=0.0022<0.01, it is concluded that the null hypothesis is rejected.

Hence We you conclude that the type of water influences the effectiveness of the detergent.

ANOVA TABLE

ANOVA
observation
	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	98.667	2	49.333	13.059	.002
Within Groups	34.000	9	3.778
Total	132.667	11