Question

Do streams with vegetated buffers (natural vegetation growth along the stream banks) have lower total phosphorus concentrations [TP] than streams without vegetated stream buffers (and if so, by what magnitude)?   We have randomly sampled 20 streams in the piedmont of North Carolina (10 with buffers and ten without buffers) and measured TP concentrations (mg/L)

Table 1. Summary statistics of the natural log of total phosphorus concentrations
	Mean (mg/L)	Standard Deviation (mg/L)	N
Buffered	1.5	0.30	10
Not Buffered	1.7	0.80	10

1. Establish appropriate hypotheses for the analysis. What type of test should you conduct?
2. Calculate the appropriate test statistic and p-value for your hypothesis test.   Interpret your p-value in a sentence.

c. What are the assumptions underlying this t-test?

Answer 1

a) H₀:

    H₁:

We should conduct a two-sample t-test.

b) The test statistic t = ()/sqrt(s1^2/n1 + s2^2/n2)

                                = (1.5 - 1.7)/sqrt((0.3)^2/10 + (0.8)^2/10)

                                = -0.74

df = (s1^2/n1 + s2^2/n2)^2/((s1^2/n1)^2/(n1 - 1) + (s2^2/n2)^2/(n2 - 1)

    = ((0.3)^2/10 + (0.8)^2/10)^2/(((0.3)^2/10)^2/9 + ((0.8)^2/10)^2/9)

    = 11

P-value = P(T < -0.74)

             = 0.2374

For alpha = 0.05, since the P-value is greater than the significance level(0.2374 > 0.05), so we should not reject the null hypothesis.

c) We should assume that the populations from which the samples are drawn must be normally distributed.