Question

Manufacturer          
A   B   C   D
25   23   25   27
23   21   25   26
21   23   25   27
23   24   21   26
To test whether the mean time needed to mix a batch of material is the same for machines produced by 4 manufacturers, the Jacobs Chemical Company obtained the following data on the time (in minutes) needed to mix the material.

What is the p value?

Does the data provide strong evidence evidence against Ho or weak evidence?

Are the mixing machine considered all equal?

Answer 1

Define the null and the alternate hypothesis as follows:

The average mixing time is different in atleast one of the 4 machines.

Let denote the number of samples. In this case, .

Let denote the number of observations in the sample respectively. In this case each.

The total sample size, , is:

The value of the test statistic, say , is found using the formula:

where, denotes the mean sum of squares due to the classes and is calculated using the formula:

,

And, denotes the mean sum of square due to errors and is calculated using the formula:

Here, denotes the samples

is the observation in the sample

is the number of observation in the sample

is the mean of the sample and  

denotes the grand mean of all the observations, and,

The table showing the necessary calculations is shown below:

	A	B	C	D
	25	23	25	27	4	0.0625	1	0.25
	23	21	25	26	0	3.0625	1	0.25
	21	23	25	27	4	0.0625	1	0.25
	23	24	21	26	0	1.5625	9	0.25
Sum	92	91	96	106	8	4.75	12	1
Mean	23	22.75	24	26.5

The grand mean is

Now,

And

Therefore,

The test statistic follows the F-distribution with degrees of freedom.

In this case, the test statistic, follows

The p-value is computed as the probability that the value the F-distribution exceeds the value of the test statistic, that is:

The value of the right tail of the F-distribution is found using the command "=F.DIST.RT()" in MS-Excel as shown below:

This implies .

Therefore, .

Assume

Since, , hence, there exists a sufficient evidence to reject the null hypothesis at 5% level of significance. Hence, it can be concluded that the average mixing time of atleast one machine is different from the mean time of another machines at a significance level of 0.05.

In case, the level of significance is assumed

Since, , hence, there exists an insufficient evidence to reject the null hypothesis at 1% level of significance. Hence, it can be concluded that the average mixing time of all the machine's is equal at a significance level of 0.01.