Learn by Doing

Matched Pairs: In this lab you will learn howto conduct a matched pairs T-test for a population mean usingStatCrunch. We will work with a data set that has historicalimportance in the development of the T-test.

Paired T hypothesis test:

μ_D = μ₁ – μ₂ : Mean of thedifference between Regular seed and Kiln-dried seed
H₀ : μ_D = 0
H_A : μ_D > 0
Hypothesis test results:

Difference	Mean	Std. Err.	DF	T-Stat	P-value
Regular seed – Kiln-dried seed	-33.727273	19.951346	10	-1.6904761	0.9391

Some features of this activity may not work well on a cell phoneor tablet. We highly recommend that you complete this activity on acomputer.

Here are the directions, grading rubric, and definition ofhigh-quality feedback for the Learn by Doingdiscussion board exercises.

A list of StatCrunch directions is provided at the bottom ofthis page.

Context

Gosset’s Seed Plot Data

William S. Gosset was employed by the Guinness brewing companyof Dublin. Sample sizes available for experimentation in brewingwere necessarily small. At that time, Gosset contacted a famousstatistician Karl Pearson (1857-1936) and was told that there wereno techniques for developing probability models for small datasets. Gosset studied under Pearson, and the outcome of his studywas perhaps the most famous paper in statistical literature, “TheProbable Error of a Mean” (1908), which introduced theT-distribution.

Since Gosset was employed by Guinness, any work he producedwould be owned by Guinness, so he published under a pseudonym,”Student”; hence, the T-distribution is often referred to asStudent’s T-distribution.

To illustrate his analysis, Gosset used the results of seeding11 different plots of land with two different types of seed:regular and kiln-dried. He wanted to determine if drying seedsbefore planting increased plant yield. Since different plots ofsoil may be naturally more fertile, this confounding variable waseliminated by using the matched pairs design and planting bothtypes of seed in all 11 plots.

The resulting data (corn yield in pounds per acre) are asfollows.

Plot	Regular seed	Kiln-dried Seed
1	1903	2009
2	1935	1915
3	1910	2011
4	2496	2463
5	2108	2180
6	1961	1925
7	2060	2122
8	1444	1482
9	1612	1542
10	1316	1443
11	1511	1535

We use these data to test the hypothesis that kiln-dried seedyields more corn than regular seed.

Because of the nature of the experimental design (matchedpairs), we are testing the difference in yield.

Plot	Regular seed	Kiln-dried Seed	Difference
1	1903	2009	–106
2	1935	1915	20
3	1910	2011	–101
4	2496	2463	33
5	2108	2180	–72
6	1961	1925	36
7	2060	2122	–62
8	1444	1482	–38
9	1612	1542	70
10	1316	1443	–127
11	1511	1535	–24

Note that the differences were calculated:regular −kiln-dried.

Variables

Regular seed: regular seeds that were traditionallyused for planting
kiln-dried: seed that were kiln-dried before planting

Data

Download the seed (Links to an external site.) datafile, and then upload the file into StatCrunch.

Prompt

1. State the hypotheses and define the parameter.
2. Checking conditions: Since Gosset invented the T-distribution,we will assume that his sample meets the conditions and proceedwith the T-test. Regardless, answer these questions to demonstrateyour understanding of the conditions for use of the T-model.

   But first you will need to review the dotplots for the data (opensin a new tab).

   1. Which graph is used to check conditions? Why?
   2. What do we look for in the graph to verify that conditions aremet?
   3. What else do we need to know about the sample of seeds beforeusing the T-test?
3. Use StatCrunch to find the T-score and the P-value. Hint: asyou work through the StatCrunch directions, keep in mind that wewant to calculate the differences asregular −kiln-dried . So you will chooseRegular seed for Sample 1 and kiln-dried seed forSample 2. (directions)
   Copy and paste the information in the StatCrunch output window intoyour initial post.
4. State a conclusion based on the context of this scenario.

EXAMPLE TO RIGHT ANSWER

1. Ho: μ=0

Ha: μ>0

The average difference is -33.73

2. a) We use the graph of the differences because that is whatwe are analyzing.

b) We look to see if the graph is normally distributed, notskewed, and doesn’t have outliers.

c) We don’t know if the data is randomly selected.

3.

Paired T hypothesis test:

μ_D = μ₁ – μ₂ : Mean of thedifference between Regular seed and Kiln-dried seed
H₀ : μ_D = 0
H_A : μ_D > 0
Hypothesis test results:

Difference	Mean	Std. Err.	DF	T-Stat	P-value
Regular seed – Kiln-dried seed	-33.727273	19.951346	10	-1.6904761	0.9391

Differences stored in column, Differences.

4. Based on the P-value of 0.9391, we do not have enoughevidence to reject the null hypothesis. There is no statisticallysignificant evidence to show that kiln-dried seeds yield more thanregular seeds.