Question

Rothamsted Experimental Station (England) has studied wheat production since 1852. Each year, many small plots of equal size but different soil/fertilizer conditions are planted with wheat. At the end of the growing season, the yield (in pounds) of the wheat on the plot is measured. For a random sample of years, one plot gave the following annual wheat production (in pounds).

3.81	4.11	3.96	4.26	3.78	3.79	4.09	4.42
3.89	3.87	4.12	3.09	4.86	2.90	5.01	3.39

Use a calculator to verify that, for this plot, the sample variance is s² ≈ 0.305.

Another random sample of years for a second plot gave the following annual wheat production (in pounds).

3.70	3.73	4.00	3.43	3.52	3.72	4.13	4.01
3.59	4.29	3.78	3.19	3.84	3.91	3.66	4.35

Use a calculator to verify that the sample variance for this plot is s² ≈ 0.095.

Test the claim that the population variance of annual wheat production for the first plot is larger than that for the second plot. Use a 1% level of significance.

(a) What is the level of significance?

State the null and alternate hypotheses.

H_o: σ₁² = σ₂²; H₁: σ₁² > σ₂²

H_o: σ₁² > σ₂²; H₁: σ₁² = σ₂²    

H_o: σ₂² = σ₁²; H₁: σ₂² > σ₁²

H_o: σ₁² = σ₂²; H₁: σ₁² ≠ σ₂²

(b) Find the value of the sample F statistic. (Use 2 decimal places.)

What are the degrees of freedom?

dfN
dfD

What assumptions are you making about the original distribution?

The populations follow dependent normal distributions. We have random samples from each population.

The populations follow independent normal distributions. We have random samples from each population.    

The populations follow independent normal distributions.

The populations follow independent chi-square distributions. We have random samples from each population.

(c) Find or estimate the P-value of the sample test statistic. (Use 4 decimal places.)

p-value > 0.100

0.050 < p-value < 0.100    

0.025 < p-value < 0.050

0.010 < p-value < 0.025

0.001 < p-value < 0.010

p-value < 0.001

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.    

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

(e) Interpret your conclusion in the context of the application.

Fail to reject the null hypothesis, there is sufficient evidence that the variance in annual wheat production is greater in the first plot.

Reject the null hypothesis, there is insufficient evidence that the variance in annual wheat production is greater in the first plot.    

Reject the null hypothesis, there is sufficient evidence that the variance in annual wheat production is greater in the first plot.

Fail to reject the null hypothesis, there is insufficient evidence that the variance in annual wheat production is greater in the first plot.

Answer 1

Solution:

(a) What is the level of significance?

Level of significance = 1% or α = 0.01

State the null and alternate hypotheses.

H_o: σ₁² = σ₂²; H₁: σ₁² > σ₂²

(b) Find the value of the sample F statistic.

F = S1^2/S2^2 = 0.305/0.095 = 3.210526316

F = 3.21

What are the degrees of freedom?

df numerator = n1 – 1 = 16 – 1 = 15

df denominator = n2 – 1 = 16 – 1 = 15

dfN	15
dfD	15

What assumptions are you making about the original distribution?

The populations follow independent normal distributions. We have random samples from each population.    

(c) Find or estimate the P-value of the sample test statistic.

P-value = 0.0152

(by using F-table or excel)

0.010 < p-value < 0.025

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

P-value = 0.0152 > α = 0.01

So, we do not reject the null hypothesis

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

Fail to reject the null hypothesis, there is insufficient evidence that the variance in annual wheat production is greater in the first plot.