Question

Phosphate pollution of lakes and rivers is a considerable problem. Elevated phosphate levels can cause algal blooms and rapid eutrophication of freshwater ecosystems. To protect these ecosystems several strict pollution laws were enacted. The phosphate levels in a river were measured at several randomly chosen sites downstream from a chemical plant before and after these laws were passed. The following measurements were recorded in µg/liter units.

Was there a significant decrease in phosphate levels after the laws were enacted? Assume the phosphate values are normally distributed.

	Before	After
n	10	12
Mean	650	500
Summation of [X1 minus X-bar] square	5,760	19,008
square of sample standard deviation	640	1,728

Suppose that you mistakenly believed that the data sets were independent samples taken from two different populations: after the law was passed the samples were taken from a different river than that of the sampling before the law was passed.

Which type of t-test would now be appropriate?

[i]: Carry this test out and determine whether you would have found that phosphate levels to be significantly higher.

[ii]: Explain the cause of any differences in your results when compared to the previous problem.

Answer 1

A paired t-test is used to compare two population means where you have two samples in which observations in one sample can be paired with observations in the other sample. Ex.. Before-and-after observations on the same subjects . In this example we used following test.

Independent (or unpaired two sample) t-test is used to compare the means of two unrelated groups of samples.

Null HypothesisThe null hypothesis assumes that the true mean difference between the paired samples is zero  Conversely, the alternative hypothesis assumes that the true mean difference between the paired samples is not equal.

Independent t-test formula:
Let A and B represent the two groups to compare.
Let mA and mB represent the means of groups A and B, respectively.
Let nA and nB represent the sizes of group A and B, respectively.
Given that, mA= 650, mB=500 , nA=10, nB= 12;

Sum(x-mA) ^2= 5760, sum(x - mB) ^2= 19008,

To protect these ecosystems several strict pollution laws were enacted. The phosphate levels in a river were measured at several randomly and independently chosen sites downstream from a chemical plant before and after these laws were passed. So that's why there is a significant decrease in phosphate level .