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 the following questions and show your work!

1.) Compute the sample mean and sample standard deviation s 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 x₂. 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’st) 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

1)

field 1

n1 = 72

Xbar1 = 11.4111

s1 = 2.0911

field 2

n2 = 80

Xbar2 = 10.6563

s2 = 3.0259

2)

df =n1+n2-2 = 150

t = 1.976

The 95% confidence interval is −0.088<μ1​−μ2​<1.597.

3)

Interval contains numbers of different signs

hence we fail to reject the null hypothesis

there is not significant difference between them

4)

we used t-distribution

No,we don't need information about the original soil water content distributions because sample size is large enough to apply central limit theorem