Question

A manufacturing company is measuring the diameter of a ball bearing in mm by 12 inspectors, each using two different kinds of calipers to test the difference between the sample means of the two calipers used. Data is shown below. Use excel to resolve.

a) Use t-test to check if there is a significant difference between the means of the population of measurements from which the two samples were selected? Use α = 0.01, 0.05, 0.1 and comment on the results.

b) Find the P-value for the test in part (a).

Inspector	Caliper 1	Caliper 2
1	0.473	0.518
2	0.512	0.552
3	0.518	0.545
4	0.492	0.521
5	0.484	0.511
6	0.512	0.492
7	0.513	0.558
8	0.536	0.545
9	0.481	0.5
10	0.533	0.575
11	0.536	0.554
12	0.538	0.515

Answer 1

a)Using Excel<data<megastat<hypothesis test< compare two groups

For  α =0.05

The output is as follows:

Hypothesis Test: Independent Groups (t-test, pooled variance)
	Caliper 1	Caliper 2
	0.51067	0.53217	mean
	0.02326	0.02597	std. dev.
	12	12	n
		22	df
		-0.021500	difference (Caliper 1 - Caliper 2)
		0.000608	pooled variance
		0.024651	pooled std. dev.
		0.010064	standard error of difference
		0	hypothesized difference
		-2.136	t
		.0440	p-value (two-tailed)

Similarly for α = 0.01

The output is as follows:

Hypothesis Test: Independent Groups (t-test, pooled variance)
	Caliper 1	Caliper 2
	0.51067	0.53217	mean
	0.02326	0.02597	std. dev.
	12	12	n
		22	df
		-0.021500	difference (Caliper 1 - Caliper 2)
		0.000608	pooled variance
		0.024651	pooled std. dev.
		0.010064	standard error of difference
		0	hypothesized difference
		-2.136	t
		.0440	p-value (two-tailed)

For  α = 0.10

Hypothesis Test: Independent Groups (t-test, pooled variance)
	Caliper 1	Caliper 2
	0.51067	0.53217	mean
	0.02326	0.02597	std. dev.
	12	12	n
		22	df
		-0.021500	difference (Caliper 1 - Caliper 2)
		0.000608	pooled variance
		0.024651	pooled std. dev.
		0.010064	standard error of difference
		0	hypothesized difference
		-2.136	t
		.0440	p-value (two-tailed)

Consider

Since this is a two-tailed test.

T statistics is -2.136

1. At alpha = 0.01, Since the p-value is greater than alpha, we fail to reject H0 and conclude that there is not sufficient evidence to prove that there is a significant difference between the means of the population of measurements from which the two samples were selected.
   

At alpha = 0.05, 0.1, Since the p-value is smaller than alpha, we reject H0 and conclude that there is sufficient evidence to suggest that there is a significant difference between the means of the population of measurements from which the two samples were selected.

b) The p-value is 0.044 in all the three cases.

Please do the comment for any doubt or clarification. Please upvote if this helps you out. Thank You!