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). 3.63 4.14 4.29 3.84 4.23 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 ≈ 0.317. Another random sample of years for a second plot gave the following annual wheat production (in pounds). 3.73 3.40 4.03 3.76 3.49 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 ≈ 0.098. 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 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

using minitab>stat>basic stat>two variances

we have

Test and CI for Two Variances: first plot, 2nd plot

Method

Null hypothesis σ(first plot) / σ(2nd plot) = 1
Alternative hypothesis σ(first plot) / σ(2nd plot) > 1
Significance level α = 0.05

F method was used. This method is accurate for normal data only.

Statistics

95% Lower
Bound for
Variable N StDev Variance StDevs
first plot 16 0.563 0.317 0.436
2nd plot 16 0.313 0.098 0.242

Ratio of standard deviations = 1.799
Ratio of variances = 3.237

Tests

Test
Method DF1 DF2 Statistic P-Value
F 15 15 3.24 0.015

(a) the level of significance is 0.01

the null and alternative hypothesis is

Ho: σ₁² = σ22; H1: σ₁² > σ₂²

(b) the value of the sample F statistic is 3.24

the degrees of freedom are  df_N = 15 ,df_D = 15

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

c )the P-value of the sample test statistic = 0.0146

0.010 < p-value < 0.025

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 sufficient evidence that the variance in annual wheat production is greater in the first plot.