Question

I would appreciate if someone could work this out and also explain how to find the upper and lower limits.

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). 3.96 3.63 3.87 3.93 3.96 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 ≈ 0.301. Another random sample of years for a second plot gave the following annual wheat production (in pounds). 4.09 3.94 3.85 3.40 3.82 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 ≈ 0.092. 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. 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. We have random samples from each population. The populations follow independent chi-square distributions. We have random samples from each population. The populations follow independent normal distributions. The populations follow dependent normal 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)

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

The populations follow independent chi-square distributions.

Formulate an analysis plan. For this analysis, the significance level is 0.01.

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

DF₁ = n₁ - 1 = 16 -1

D.F₁ = 15

DF₂ = n₂ - 1 = 16 -1

D.F₂ = 15

Test statistics:-

F = 3.262

c)

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

p value for the F distribution = 0.014.

0.010 < p-value < 0.025

Interpret results. Since the P-value (0.014) is greater than the significance level (0.01), we have to accept the null hypothesis.

d) At the α = 0.01 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 is greater in the first plot.