Question

Two plots at Rothamsted Experimental Station were studied for production of wheat straw. For a random sample of years, the annual wheat straw production (in pounds) from one plot was as follows. 6.40 7.03 6.47 5.70 7.31 7.18 7.06 5.79 6.24 5.91 6.14 Use a calculator to verify that, for this plot, the sample variance is s2 ≈ 0.341. Another random sample of years for a second plot gave the following annual wheat production (in pounds). 6.40 7.10 5.91 5.84 7.22 5.58 5.47 5.86 Use a calculator to verify that the sample variance for this plot is s2 ≈ 0.447. Test the claim that there is a difference (either way) in the population variance of wheat straw production for these two plots. Use a 5% level of signifcance. (a) What is the level of significance? State the null and alternate hypotheses. Ho: σ12 = σ22; H1: σ12 > σ22 Ho: σ12 > σ22; H1: σ12 = σ22 Ho: σ22 = σ12; H1: σ22 > σ12 Ho: 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 independent normal distributions. The populations follow independent normal distributions. We have random samples from each population. The populations follow dependent normal distributions. We have random samples from each population. 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 > 0.200 0.100 < p-value < 0.200 0.050 < p-value < 0.100 0.020 < p-value < 0.050 0.002 < p-value < 0.020 p-value < 0.002 (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 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 differs between the two plots. Reject the null hypothesis, there is insufficient evidence that the variance in annual wheat production differs between the two plots. Reject the null hypothesis, there is sufficient evidence that the variance in annual wheat production differs between the two plots. Fail to reject the null hypothesis, there is insufficient evidence that the variance in annual wheat production differs between the two plots.

Answer 1

Solution:-

a) The level of significance is 0.05.

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis H₀: σ₁² = σ₂²

Alternative hypothesis H_A: σ₁² σ₂²

Formulate an analysis plan. For this analysis, the significance level is 0.05.

Analyze sample data. Using sample data, the degrees of freedom (DF), and the test statistic (F).

b)

Test statistics:-

F = 0.5820

DF₁ = n₁ - 1 = 11 -1

D.F₁ = 10

DF₂ = n₂ - 1 = 8 -1

D.F₂ = 7

We have random samples from each population.

Since the first sample had the larger standard deviation, this is a right-tailed test.

c)

p value for the F distribution = 0.7890

p-value > 0.200

Interpret results. Since the P-value (0.789) is greater than the significance level (0.05), hence we failed to reject the null hypothesis.

d) Based on your answers in parts (a) to (c), fail to reject the null hypothesis.

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Fail to reject the null hypothesis, there is insufficient evidence that the variance in annual wheat production differs between the two plots.