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).
4.29 | 3.90 | 3.99 | 3.87 | 4.29 | 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 ≈ 0.309.
Another random sample of years for a second plot gave the following
annual wheat production (in pounds).
3.58 | 3.70 | 3.85 | 3.40 | 3.79 | 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 > σ22Ho: σ12 > σ22; H1: σ12 = σ22 Ho: σ22 = σ12; H1: σ22 > σ12Ho: 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 chi-square 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.
(c) Find or estimate the P-value of the sample test
statistic. (Use 4 decimal places.)
p-value > 0.1000.050 < p-value < 0.100 0.025 < p-value < 0.0500.010 < p-value < 0.0250.001 < p-value < 0.010p-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.
Variance Calculation First Plot:
Thus, s12 ≈ 0.309
Variance Calculation Second Plot:
Thus, s22 ≈ 0.092
Answer a)
The level of significance = 0.01
The null and alternate hypotheses are as follows:
Ho: σ12 = σ22;
H1: σ12 > σ22
Here, σ12 = population variance of annual wheat production for the first plot and σ2 = population variance of annual wheat production for the second plot
Answer b)
Test Statistics
The value of the sample F statistic is 3.36
The degrees of freedom are
dfN = n1 - 1 = 16 - 1 = 15
dfD = n2 - 1 = 16 - 1 = 15
Assumptions
We have random samples from each population.
The populations follow independent normal distributions.
Answer c)
The p-value corresponding to F(15,15) = 3.359 is 0.0124 (Obtained using online calculator. Screenshot attached)
Thus, correct option is 0.010 < p-value < 0.025
Answer d)
Since p-value (0.0124) > α = 0.01, we fail to reject null hypothesis.
Thus,
At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant
Answer e)
Interpretation
Fail to reject the null hypothesis, there is insufficient evidence that the variance in annual wheat production is greater in the first plot.
Rothamsted Experimental Station (England) has studied wheat production since 1852. Each year, many small plots of...
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