Question

The U.S. Department of Agriculture (USDA) uses sample surveys to produce important economic estimates. One pilot study estimated durum wheat prices in July and in January using independent samples of wheat producers in the two months.

The data is given below.

Price	Month
6.2335	July
6.3215	July
6.4457	July
6.1488	July
6.3079	July
5.8976	January
6.2064	January
6.1278	January
6.0468	January
5.9692	January

Although there is variation among prices within each month, the top four prices are all from July and the four lowest prices are from January. You will run a significance test to check that if the difference between months is just by chance. You’ll use α = 0.05 for significance test. Assume that the two Normal population distributions have the same standard deviation.

1. 1. Write out the null and alternative hypothesis associated with the research question. (2p)
2. 2. What type of statistical test will you use to answer the proposed research question? (Note: Is this a z-test or a t-test? (2p)
3. 3. What is the critical value at the 0.05 level of significance? Be sure and include whether this critical value is a z or t value and, if appropriate, include the degrees of freedom associated with this statistical test. (4p)
4. 4. What is the calculated value of the test statistic? (4p)
   5. What decision should be made about the null hypothesis? In other words, should you reject or retain the null hypothesis? (3p)
   6. Provide a brief conclusion regarding your findings. Use your powerpoint lecture slides for writing out the interpretation of your results. (5p)
   7. Show your Excel output (10p)

Answer 1

Solution

Final answers are given below. Details are given under Part (g).

Let X = July price

      Y = January price

Given, X ~ N(µ₁, σ₁²) and Y ~ N(µ₂, σ₂²), where σ₁² = σ₂² = σ², say and σ² is unknown.

Part (a)

Hypotheses

Null: H₀: µ1 = µ2 Vs Alternative: H_A: µ1 ≠ µ2 Answer 1

Part (b)

Type of test

t-test [because population variance is unknown] Answer 2

Part (c)

Critical value

t₈, _0.025 = 2.306 Answer 3

Part (d)

Calculated value of the test statistic

3.2777   Answer 4

[Test statistic =

t = (Xbar - Ybar)/{s√(2/n)} where

s² = (s₁² + s₂²)/2;

Xbar and Ybar are sample averages and s₁,s₂ are sample standard deviations based on n observations each on X and Y.

Part (e)

Decision

Reject null hypothesis Answer 5

[Because tcal > tcrit]

Part (f)

Conclusion

Mean price in July is different from the price in January Answer 6

Part (g)

Excel output

n	5
Xbar	6.29
Ybar	6.05
s1	0.1103
s2	0.1228
s^2	0.013619
s	0.1167
|tcal|	3.2777
α	0.05
tcrit	2.306004

Answer 7

DONE