Question

In an experiment to determine whether there is a systematic difference between the weights obtained with two different mass balances, six specimens were weighed, in grams, on each balance. The following data were obtained:

Specimen A B
1 13.76 13.74
2 12.47 12.45
3 10.09 10.08
4 8.91 8.92
5 13.57 13.54
6 12.74 12.75
Can you conclude that the mean weight differs between the two balances?
i). State the null and alternative hypotheses.
ii). Compute the test statistic.
iii). State a conclusion using the a =0.05 level of significance.

Answer 1

H0: μd=0 Ha: μd≠0

t= 0.07607

On the basis of this we conclude that the mean weight differs between the two balances.

Explanation:

The null and alternative hypotheses as

H0: μd=0 Ha: μd≠0

Significance level is set at ∝= 0.05

The critical region is t ( base alpha by 2 with df=5) ≥ ± 2.571

The test statistic under H0 is

t = d/ sd/ √n

Which has t distribution with n-1 degrees of freedom

Specimen A B d = a - b d²

1 13.76 13.74 0.02 0.004

2 12.47 12.45 0.02 0.004

3 10.09 10.08 0.01 0.001

4 8.91 8.92 -0.01 0.001

5 13.57 13.54 0.03 0.009

6 12.74 12.75 -0.01 0.001

∑ 0.06 0.0173

d`= ∑d/n= 0.006/6= 0.001

sd²= 1/6( 0.0173- 0.006²/6) = 1/6 ( 0.017294) = 0.002882

sd= 0.05368

t= 0.001/ 0.05368/ √6

t= 0.18629/2.449

t= 0.07607

Since the calculated value of t= 0.07607 does not falls in the rejection region we therefore accept the null hypothesis at 5 % significance level . On the basis of this we conclude that the mean weight differs between the two balances.