Question

Are yields for organic farming different from conventional farming yields? Independent random samples from method A (organic farming) and method B (conventional farming) gave the following information about yield of lima beans (in tons/acre). (Reference: Agricultural Statistics, United States Department of Agriculture.)

Method A	1.16	1.45	1.17	1.54	1.91	1.26	1.20	1.95	1.64	1.44	1.81
Method B	1.15	2.17	2.10	1.78	1.90	2.06	1.14	2.08	1.41	2.03	1.30	1.42

Use a 1% level of significance to test the hypothesis that there is no difference between the yield distributions.

(a) What is the level of significance?

State the null and alternate hypotheses.

H_o: Distributions are the same. H₁: Distributions are different.

H_o: Distributions are the same. H₁: Distributions are the same.   

H_o: Distributions are different. H₁: Distributions are different.

H_o: Distributions are different. H₁: Distributions are the same.

(b) Compute the sample test statistic. (Use 2 decimal places.)

What sampling distribution will you use?

normal

Student's t  

  chi-square

uniform

What conditions are necessary to use this distributions?

Both sample sizes must be less than 10.

Both sample sizes must be greater than 10.    

At least one sample size must be less than 10.

At least one sample size must be greater than 10.

(c) Find the P-value of the sample test statistic. (Use 4 decimal places.)

(d) Conclude the test?

At the ? = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the ? = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.  

  At the ? = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the ? = 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.

Reject the null hypothesis, there is sufficient evidence that the yield distributions are different.

Fail to reject the null hypothesis, there is sufficient evidence that the yield distributions are different.    

Fail to reject the null hypothesis, there is insufficient evidence that the yield distributions are different.

Reject the null hypothesis, there is insufficient evidence that the yield distributions are different.

Answer 1

a)

The null and alternate hypotheses are:

H₀: Distributions are the same. H₁: Distributions are different.

b)Test statistic

First,We will arrange the two methods jointly in ascending order and rank the two groups separately .

Then,we will add up the rank of the group with the smaller sample size,in this case,it is method A.

The sum of the ranks is denoted by R:

R=114

Let n₁ be the size of the smaller sample size,i.e., n₁=11 and

n₂ be the size of the larger sample size,i.e., n₂=12

Then,

Test statistic is

Test statistic is z=-1.11

Method A	Rank	Method B	Rank
1.16	3	1.14	1
1.17	4	1.15	2
1.2	5	1.3	7
1.26	6	1.41	8
1.44	10	1.42	9
1.45	11	1.78	14
1.54	12	1.9	16
1.64	13	2.03	19
1.81	15	2.06	20
1.91	17	2.08	21
1.95	18	2.1	22
		2.17	23
Sum	114		162

We will use Normal distribution

Both sample sizes must be greater than 10.   

c) we can find the exact p-value in excel by using the condition

p-value=2*(1-NORM.S.DIST(1.11,TRUE))

p-value=0.2670

Note :we are multiplying by 2 as it is a two-tailed test.

If p-value is greater than alpha vlaue,then we do not reject H_0.

Here our p-value=0.2670>=0.01,hence we do not reject H_0.

d) At the = 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 yield distributions are different.