Local environmentalists want to test if there is a difference in the pH of three different streams (called A, B, and C) feeding
1 answer:
Answer:
Step-by-step 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=μ
=μ
=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=μ
H₁: μ≠μ
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=μ
H₁: μ≠μ
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=μ
H₁: μ≠μ
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!
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Answer: d. x² + 6x +8
Answer: d. x² + 6x +8
Prove:
____________________________________________
Base Step: For n=1:
and
--------------------------------------------------------------------------
Induction Hypothesis: Assume true for n=k. Meaning:
assumed to be true.
--------------------------------------------------------------------------
Induction Step: For n=k+1:
by our Induction Hypothesis, we can replace every term in this summation (except the last term) with the right hand side of our assumption.
From here, think about what you are trying to end up with.
For n=k+1, we WANT the formula to look like this:
That thing on the right hand side is what we're trying to end up with. So we need to do some clever Algebra.
Combine the (k+1) and 1/2, put the 2 in the bottom,
We want to end up with a 2^k as our final denominator, so our middle term is missing a power of 2. Let's multiply top and bottom by 2,
Distribute the -2 and combine the fractions together,
Combine like-terms,
pull the negative back out,
And ta-da! We've done it!
We can break apart the +3 into +1 and +2,
and the +0 in the bottom can be written as -1 and +1,
____________________________________________
Base Step: For n=1:
and
--------------------------------------------------------------------------
Induction Hypothesis: Assume true for n=k. Meaning:
assumed to be true.
--------------------------------------------------------------------------
Induction Step: For n=k+1:
by our Induction Hypothesis, we can replace every term in this summation (except the last term) with the right hand side of our assumption.
From here, think about what you are trying to end up with.
For n=k+1, we WANT the formula to look like this:
That thing on the right hand side is what we're trying to end up with. So we need to do some clever Algebra.
Combine the (k+1) and 1/2, put the 2 in the bottom,
We want to end up with a 2^k as our final denominator, so our middle term is missing a power of 2. Let's multiply top and bottom by 2,
Distribute the -2 and combine the fractions together,
Combine like-terms,
pull the negative back out,
And ta-da! We've done it!
We can break apart the +3 into +1 and +2,
and the +0 in the bottom can be written as -1 and +1,