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.

6.47 5.70 6.75 6.47
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 ≈ 0.316.

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

8.08 6.96 6.89 6.05
7.22 5.58 5.47 5.86

Use a calculator to verify that the sample variance for this plot
is s2 ≈ 0.842.

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.

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.

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 > 0.200

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

a)level of significance =0.05

Ho: σ12 = σ22; H1: σ12 ≠ σ22

b)

Test statistic =s_{1}^{2}/s_{2}^{2}
= |
2.66 |

dfN= | 7 |

dfD= | 10 |

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

0.100 < p-value < 0.200

d)

At the α = 0.05 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 differs between the two plots.

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.
6.75 6.26 6.96 5.77 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 ≈ 0.334.
Another random sample of years for a second plot gave the
following annual wheat production (in pounds).
8.15
5.91
6.19
7.10...

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.
6.61
6.40
5.84
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 ≈ 0.304.
Another random sample of years for a second plot gave the following
annual wheat production (in pounds).
7.87
5.70
7.24
5.98...

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.
6.96
7.10
5.84
5.91
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 ≈ 0.384.
Another random sample of years for a second plot gave the following
annual wheat production (in pounds).
5.91
5.91
5.91
5.91...

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.
6.54
7.17
5.91
7.10
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 ≈ 0.357.
Another random sample of years for a second plot gave the following
annual wheat production (in pounds).
6.12
6.89
6.61
8.15...

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.
5.98
6.12
6.05
6.96
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 ≈ 0.326.
Another random sample of years for a second plot gave the following
annual wheat production (in pounds).
7.94
8.15
7.03
6.47...

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.
6.61
7.03
5.98
6.96
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 ≈ 0.321.
Another random sample of years for a second plot gave the following
annual wheat production (in pounds).
6.47
7.03
7.24
8.15...

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.
7.10 6.19 6.12 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
≈ 0.317. Another random sample of years for a second plot gave the
following annual wheat production (in pounds). 6.54 6.26 6.47 7.52...

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.
7.03 6.54 6.12 6.75 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
≈ 0.294. Another random sample of years for a second plot gave the
following annual wheat production (in pounds). 5.91 5.77 7.45 7.66...

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.
7.10
5.91
6.19
6.75
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 ≈ 0.335.
Another random sample of years for a second plot gave the following
annual wheat production (in pounds).
7.38
6.26
7.45
5.84...

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 5.91 7.10 5.77 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 ≈ 0.374.
Another random sample of years for a second plot gave the
following annual wheat production (in pounds). 6.68 6.61 6.05 6.89...

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