Question

Local environmentalists want to test if there is a difference in the pH of three different streams (called A, B, and C) feeding into Lake Travis. They took a random sample of 15 measurements from each stream and, after confirming all relevant assumptions, ran a one-way ANOVA on the data and got a significant (p<0.05) result.

a. Write the appropriate null and alternative hypotheses for this analysis.
b. List one potential confounding variable.
c. In a post-hoc analysis, they got the following p values. Which pairs of streams have significantly different mean pH? Show work for full credit.

Stream A vs. B: p = 0.003
Stream A vs. C: p= 0.029
Stream A vs. C: p= 0.041

Answer 1

Streams A vs. B and A vs. C have significantly different mean pH, while there is no significant difference between streams B and C.

a. The appropriate null and alternative hypotheses for the one-way ANOVA are:

Null Hypothesis (H0): The mean pH is the same in streams A, B, and C.

Alternative Hypothesis (H1): At least one of the streams has a different mean pH.

b. One potential confounding variable could be the presence of industrial discharges or agricultural runoff into the streams, as these external factors might influence the pH levels independently of the natural conditions in each stream.

c. In the post-hoc analysis, the p-values are given for each pair of streams:

Stream A vs. B: p = 0.003

Stream A vs. C: p = 0.029

Stream B vs. C: p = 0.041

Comparing the p-values to the significance level (α = 0.05), the pairs with significantly different mean pH are:

Stream A vs. B (p = 0.003): Reject the null hypothesis; there is a significant difference.

Stream A vs. C (p = 0.029): Reject the null hypothesis; there is a significant difference.

Stream B vs. C (p = 0.041): Fail to reject the null hypothesis; there is no significant difference.

Answer 2

Explanation:

Hello!

The variable Y: pH of the water of a stream that feeds into Lake Travis (stream A, stream B, and stream C)

This is a one way ANOVA with one factor (ph of the water) and three levels (stream A, stream B, and stream C).

a)

The parameters of interest are the population mean of pH of each stream, then the hypotheses for this ANOVA are

H₀: μ
`_(A)`=μ
`_(B)`=μ
`_(C)`=μ

H₁: At least one population mean is not equal to the others.

b)

There are many factors that may change the pH of water, for example:

1) Polution

2) Soil components (different concentrations of minearls) and pH of the soil

c)

The null hypothesis was rejected, so they made a post-hoc analysis to see wich of the population means is different from the others:

Using the same α: 0.05 as before

The deicsion rule for the p-value approach is

If p-value ≤ α, reject the null hypothesis.

If p-value ≥ α, do not reject the null hypothesis

1.

Stream A vs. B: p = 0.003

H₀: μ
`_(A)`=μ
`_(B)`

H₁: μ
`_(A)`≠μ
`_(B)`

p-value: 0.003 < α: 0.05 ⇒ Reject null hypothesis. The average pH of stream A is different than the average pH of stream B.

2.

Stream A vs. C: p= 0.029

H₀: μ
`_(A)`=μ
`_(C)`

H₁: μ
`_(A)`≠μ
`_(C)`

p-value: 0.029 < α: 0.05 ⇒ Reject null hypothesis. The average pH of stream A is different than the average pH of stream C.

3.

Stream A vs. C: p= 0.041

H₀: μ
`_(A)`=μ
`_(C)`

H₁: μ
`_(A)`≠μ
`_(C)`

p-value: 0.041 < α: 0.05 ⇒ Reject null hypothesis. The average pH of stream A is different than the average pH of stream C.

I hope this helps!