Question

The following data represent soil water content (percentage of water by volume) for independent random samples of soil taken from two experimental fields growing bell peppers.

Soil water content from field I: x1; n1 = 72

15.2 11.3 10.1 10.8 16.6 8.3 9.1 12.3 9.1 14.3 10.7 16.1 10.2 15.2 8.9 9.5 9.6 11.3 14.0 11.3 15.6 11.2 13.8 9.0 8.4 8.2 12.0 13.9 11.6 16.0 9.6 11.4 8.4 8.0 14.1 10.9 13.2 13.8 14.6 10.2 11.5 13.1 14.7 12.5 10.2 11.8 11.0 12.7 10.3 10.8 11.0 12.6 10.8 9.6 11.5 10.6 11.7 10.1 9.7 9.7 11.2 9.8 10.3 11.9 9.7 11.3 10.4 12.0 11.0 10.7 8.5 11.1

Soil water content from field II: x2; n2 = 80

12.1 10.2 13.6 8.1 13.5 7.8 11.8 7.7 8.1 9.2 14.1 8.9 13.9 7.5 12.6 7.3 14.9 12.2 7.6 8.9 13.9 8.4 13.4 7.1 12.4 7.6 9.9 26.0 7.3 7.4 14.3 8.4 13.2 7.3 11.3 7.5 9.7 12.3 6.9 7.6 13.8 7.5 13.3 8.0 11.3 6.8 7.4 11.7 11.8 7.7 12.6 7.7 13.2 13.9 10.4 12.9 7.6 10.7 10.7 10.9 12.5 11.3 10.7 13.2 8.9 12.9 7.7 9.7 9.7 11.4 11.9 13.4 9.2 13.4 8.8 11.9 7.1 8.8 14.0 14.2

Answer questions 1 to 5 below. Show all your work with steps and not just final answers.

1. Compute the sample mean and sample standard deviations of soil water content for field I and for field II.

2. Let μ1be the population mean for x1and let μ2 be the population mean for x. Find a 95% confidence interval for μ1 – μ2.

3. Examine the confidence interval and explain what it means in the context of this problem. Does the interval consist of numbers that are all positive? all negative? of different signs? At the 95% level of confidence, is the population mean soil water content of the field I higher than that of the field II?

4. Which distribution (Standard Normal or Student’s t) did you use? Explain why? Do you need information about the original soil water content distributions?

5. Use α = 0.01 to test the claim that the population mean soil water content of field I is higher than that of field II. Please provide the following information:

(a) What is the level of significance? State the null and alternate hypotheses.

(b) What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic to nearest hundredth.

(c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

(e) Interpret your conclusion in the context of the application.

Answer 1

Q1. The mean and Standard deviation are computed as follow

X1	X2
15.2	12.1	14.36	2.08
11.3	10.2	0.01	0.21
10.1	13.6	1.72	8.67
10.8	8.1	0.37	6.53
16.6	13.5	26.92	8.09
8.3	7.8	9.68	8.16
9.1	11.8	5.34	1.31
12.3	7.7	0.79	8.74
9.1	8.1	5.34	6.53
14.3	9.2	8.35	2.12
10.7	14.1	0.51	11.86
16.1	8.9	21.99	3.08
10.2	13.9	1.47	10.52
15.2	7.5	14.36	9.96
8.9	12.6	6.31	3.78
9.5	7.3	3.65	11.26
9.6	14.9	3.28	18.01
11.3	12.2	0.01	2.38
14	7.6	6.7	9.34
11.3	8.9	0.01	3.08
15.6	13.9	17.55	10.52
11.2	8.4	0.04	5.09
13.8	13.4	5.71	7.53
9	7.1	5.81	12.65
8.4	12.4	9.07	3.04
8.2	7.6	10.31	9.34
12	9.9	0.35	0.57
13.9	26	6.19	235.43
11.6	7.3	0.04	11.26
16	7.4	21.06	10.6
9.6	14.3	3.28	13.28
11.4	8.4	0	5.09
8.4	13.2	9.07	6.47
8	7.3	11.64	11.26
14.1	11.3	7.23	0.41
10.9	7.5	0.26	9.96
13.2	9.7	3.2	0.91
13.8	12.3	5.71	2.7
14.6	6.9	10.17	14.11
10.2	7.6	1.47	9.34
11.5	13.8	0.01	9.88
13.1	7.5	2.85	9.96
14.7	13.3	10.82	6.99
12.5	8	1.19	7.06
10.2	11.3	1.47	0.41
11.8	6.8	0.15	14.87
11	7.4	0.17	10.6
12.7	11.7	1.66	1.09
10.3	11.8	1.23	1.31
10.8	7.7	0.37	8.74
11	12.6	0.17	3.78
12.6	7.7	1.41	8.74
10.8	13.2	0.37	6.47
9.6	13.9	3.28	10.52
11.5	10.4	0.01	0.07
10.6	12.9	0.66	5.03
11.7	7.6	0.08	9.34
10.1	10.7	1.72	0
9.7	10.7	2.93	0
9.7	10.9	2.93	0.06
11.2	12.5	0.04	3.4
9.8	11.3	2.6	0.41
10.3	10.7	1.23	0
11.9	13.2	0.24	6.47
9.7	8.9	2.93	3.08
11.3	12.9	0.01	5.03
10.4	7.7	1.02	8.74
12	9.7	0.35	0.91
11	9.7	0.17	0.91
10.7	11.4	0.51	0.55
8.5	11.9	8.47	1.55
11.1	13.4	0.1	7.53
	9.2		2.12
	13.4		7.53
	8.8		3.45
	11.9		1.55
	7.1		12.65
	8.8		3.45
	14		11.18
	14.2		12.56
= 821.6	= 852.5	= 310.48	= 723.26

2) confidence interval can be calculated as follow

where

and critical value for 95% of confidence interval

3. The 95% C.I for difference of means is (-0.073, 1.582) which mean with 95% confidence the difference will vary between -0.073 and 1.582.

Now to check at 95% level of confidence, is the population mean soil water content of the field I higher than that of the field II, which can be checked Via t.test for two means.

We will set null hypothesis

Vs   

The test statistics is

and its corresponding

Which is less then 0.05 therefore we will reject null and accept alternate hypothesis.

4. Since in practice, the two‐sample z‐test is not used often, because the two population standard deviations σ ₁ and σ ₂ are usually unknown. Instead, sample standard deviations and the t‐distribution are used. we have also used  t‐distribution. No we need not any information.

5.

a. The level of significance = 0.01

We will set null hypothesis

Vs   

b. we use t.test and its Assumption are

(i) the parent population is normal

(ii) sample are independent

(ii) homogeneity of variance

The test statistics is

c) Its corresponding

d) Which is greater then 0.01 therefore we will accept null and reject alternate hypothesis.

e) At 0.01 significance level with 99% confidence we accept our hypothesis that mean soil water content of field I is equal to field II and reject that mean soil water content of field I is higher than that of field II.