Question

Phosphorous is a chemical that is found in many household cleaning products. Unfortunately, phosphorous also finds its way into surface water, where it can harm fish, plants, and other wildlife. Two methods of phosphorous reduction are being studied. At a random sample of 7 locations, both methods were used and the total phosphorous reduction (mg/L) was recorded. Note: For degrees of freedom d.f. not in the Student's t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value a small amount and thereby produce a slightly more "conservative" answer.

Site	1	2	3	4	5	6	7
Method I:	0.013	0.020	0.015	0.055	0.007	0.002	0.010
Method II:	0.014	0.045	0.017	0.039	0.017	0.001	0.013

Do these data indicate a difference (either way) in the average reduction of phosphorous between the two methods? Use

α = 0.05.

(Let d = Method I − Method II.)

(i) What is the level of significance?

What is the value of the sample test statistic? (Round your answer to three decimal places.)

Answer 1

Here by the problem,

Phosphorous is a chemical that is found in many household cleaning products. Unfortunately, phosphorous also finds its way into surface water, where it can harm fish, plants, and other wildlife. Two methods of phosphorous reduction are being studied. At a random sample of 7 locations, both methods were used and the total phosphorous reduction (mg/L) was recorded. The data on that reduction of phosphorus by both methods are noted as follows. Now if we want to check whether these data indicate a difference (either way) in the average reduction of phosphorous between the two methods, and we assume as the mean difference of phosphorus between two methods then we want to check,

where d=method 1 - method 2,

So here the level of significance is 0.05

Now inorder to test that the sample test statistic be,

where be the sample mean and sample standard deviation of the difference of n samples and under null hypothesis, or here for n=7,

So the calculation of the test statistic is as follows,

Site	Method I	Method II	d=Method I-Method II	d-mean(d)	(d-mean(d))^2
1	0.013	0.014	-0.001	0.0024	0.00000576
2	0.02	0.045	-0.025	-0.0216	0.00046656
3	0.015	0.017	-0.002	0.0014	0.00000196
4	0.055	0.039	0.016	0.0194	0.00037636
5	0.007	0.017	-0.01	-0.0066	0.00004356
6	0.002	0.001	0.001	0.0044	0.00001936
7	0.01	0.013	-0.003	0.0004	0.00000016
Total			-0.024		0.00091372
Average			-0.0034

So clearly here,

and

So the test statistic be,

So as it is a both tailed test the p-value be,

Now as p-value=0.4935>> 0.05=level of significance hence based on data we fail to reject null hypothesis and conclude that these data do not indicate a difference (either way) in the average reduction of phosphorous between the two methods.

Hence the answer................

Thank you..............