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.

6.19	6.33	6.61	6.82	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 ≈ 0.272.

Another random sample of years for a second plot gave the following annual wheat production (in pounds).

6.12

6.82

7.80

8.15

7.22

5.58

5.47

5.86

Use a calculator to verify that the sample variance for this plot is s² ≈ 1.052.

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.

H_o: σ₁² = σ₂²; H₁: σ₁² > σ₂²

H_o: σ₁² > σ₂²; H₁: σ₁² = σ₂²    

H_o: σ₂² = σ₁²; H₁: σ₂² > σ₁²

H_o: 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 independent chi-square distributions. We have random samples from each population.

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 > 0.200

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

	Values ( X )	Σ ( Xi- X̅ )2	Values ( Y )	Σ ( Yi- Y̅ )2
	6.19	0.1007	6.12	0.2576
	6.33	0.0314	6.82	0.0371
	6.61	0.0105	7.8	1.3748
	6.82	0.0978	8.15	2.318
	7.31	0.6443	7.22	0.3511
	7.18	0.4525	5.58	1.0973
	7.06	0.3055	5.47	1.3398
	5.79	0.5145	5.86	0.5891
	6.24	0.0714
	5.91	0.3568
	6.14	0.1349
Total	71.58	2.7203	53.02	7.3648

Mean X̅ = Σ Xi / n
X̅ = 71.58 / 11 = 6.5073
Sample Standard deviation S_X = √ ( (Xi - X̅ )2 / n - 1 )
S_X = √ ( 2.7203 / 11 -1 ) = 0.5216

Sample Variance = 0.5216²= 0.272

Mean Y̅ = ΣYi / n
Y̅ = 53.02 / 8 = 6.6275
Sample Standard deviation S_Y = √ ( (Yi - Y̅ )2 / n - 1 )
S_Y = √ ( 7.3648 / 8 -1) = 1.0257

Sample Variance = 1.0257² = 1.052

Part a)

α = 0.05

H_o: = σ₂²; H₁: σ₁² ≠ σ₂²

Part b)

Test Statistic :-

f = 0.272 / 1.052
f = 0.26

Degree of freedom

n1 - 1 = 11 - 1 = 10

n2 = 8 - 1 = 7

The populations follow independent normal distributions.

Part c)

P value = 2 * P ( f > 0.2586 ) = 0.0534

0.050 < p-value < 0.100

Part d)
Reject null hypothesis if P value < α = 0.05
Since P value = 0.0534 > 0.05, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

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

Part e)

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