The ability of ecologists to identify regions of greatest sp

The ability of ecologists to identify regions of greatest species richness could have an impact on the preservation of genetic diversity, a major objective of the World Conservation Strategy. A study used a sample of n = 31 lakes to obtain the estimated regression equation

y = 3.89 + 0.033x₁ + 0.024x₂ + 0.023x₃ ? 0.0080x₄ ? 0.13x₅ ? 0.72x₆

where y = species richness, x₁ = watershed area, x₂ = shore width, x₃ = poor drainage (%), x₄ = water color (total color units), x₅ = sand (%), and x₆ = alkalinity. The coefficient of multiple determination was reported as R² = 0.82. Carry out a test of model utility.
State the appropriate hypotheses.

H₀: b₁ = b₁ = ... = b₆ = 0

H_a: at least one among b₁, ..., b₆ is not zero

H₀: b₁ not equal b₁ not equal ... not equal b₆ not equal 0

H_a: b₁ = b₁ = ... = b₆ = 0

H₀: b₁ = b₁ = ... = b₆ = 0

H_a: no b_i = 0

H₀: b₁ not equal b₁ not equal ... not equal b₆ not equal 0

H_a: at least one among b₁, ..., b₆ is zero

State the rejection region and compute the test statistic value. Use ? = 0.05. Round your answers to two decimal places.

State the conclusion in the problem context.

Reject H₀. The model is judged not useful.

Reject H₀. The model is judged useful.    

Fail to reject H₀. The model is judged useful.

Fail to reject H₀. The model is judged not useful.

You may need to use the appropriate table in the Appendix of Tables to answer this question.

Solution

the estimated regression equation

y = 3.89 + 0.033x₁ + 0.024x₂ + 0.023x₃ - 0.0080x₄ - 0.13x₅ - 0.72x₆

The hypothesis is

H₀: b₁ = b₁ = ... = b₆ = 0

versus
H_a: at least one among b₁, ..., b₆ is not zero

The test statistic value and correlation coefficient is

f = 18.22 and R² = 0.82

The f critical value from the distribution table with degrees of freedom n-2=31-2=29 at 0.05(two sided) significance level is 2.045

Because f-statistics is greter than critical value 2.045 we reject the null hypothesis.we conclude The model is judged useful. there exist a correlation. rejection area -2.045 to 2.045 if the statistics value fall outside this interval we reject the null hypothesis.