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).
3.87 | 4.08 | 3.60 | 3.84 | 3.81 | 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 s^{2} ≈ 0.305.
Another random sample of years for a second plot gave the following
annual wheat production (in pounds).
3.73 | 3.46 | 4.03 | 3.43 | 3.97 | 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 s^{2} ≈ 0.099.
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}: σ_{1}^{2} = σ_{2}^{2}; H_{1}: σ_{1}^{2} > σ_{2}^{2}
H_{o}: σ_{1}^{2} > σ_{2}^{2}; H_{1}: σ_{1}^{2} = σ_{2}^{2}
H_{o}: σ_{2}^{2} = σ_{1}^{2}; H_{1}: σ_{2}^{2} > σ_{1}^{2}
H_{o}: σ_{1}^{2} = σ_{2}^{2}; H_{1}: σ_{1}^{2} ≠ σ_{2}^{2}
(b) Find the value of the sample F statistic. (Use 2
decimal places.)
What are the degrees of freedom?
df_{N} | |
df_{D} |
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.
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.
(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.
The statistical software output for this problem is:
Hence,
a) Level of significance = 0.01
Hypotheses: H_{o}: σ_{1}^{2} = σ_{2}^{2}; H_{1}: σ_{1}^{2} > σ_{2}^{2}
b) Test statistic = 3.08
dfN = 15
dfD = 15
The populations follow independent normal distributions. We have random samples from each population.
c) 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 insufficient evidence that the variance in annual wheat production is greater in the first plot.
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