William S. Gosset was employed by the Guinness brewing company of Dublin. Sample sizes available for experimentation in brewing were necessarily small. At that time, Gosset contacted a famous statistician Karl Pearson (1857-1936) and was told that there were no techniques for developing probability models for small data sets. Gosset studied under Pearson, and the outcome of his study was perhaps the most famous paper in statistical literature, "The Probable Error of a Mean" (1908), which introduced the T-distribution. Since Gosset was employed by Guinness, any work he produced would be owned by Guinness, so he published under a pseudonym, "Student"; hence, the T-distribution is often referred to as Student's T-distribution.
To illustrate his analysis, Gosset used the results of seeding 11 different plots of land with two different types of seed: regular and kiln-dried. He wanted to determine if drying seeds before planting increased plant yield. Since different plots of soil may be naturally more fertile, this confounding variable was eliminated by using the matched pairs design and planting both types of seed in all 11 plots.
The resulting data (corn yield in pounds per acre) are as follows.
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 seed yields more corn than regular seed.
Because of the nature of the experimental design (matched pairs), we are testing the difference in yield.
Plot Regular seed Kiln-dried Seed Difference
1 1903 1609 –106
2 1935 1915 20
3 1910 1611 –101
4 2496 2463 33
5 2108 2180 –72
6 1961 1925 36
7 1660 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.
Prompt
State the hypotheses and define the parameter.
Checking conditions: Since Gosset invented the T-distribution, we will assume that his sample meets the conditions and proceed with the T-test. Regardless, answer these questions to demonstrate your understanding of the conditions for use of the T-model:
Click here to see some dotplots of the data.
–Which graph is used to check conditions? Why?
–What do we look for in the graph to verify that conditions are met?
–What else do we need to know about the sample of seeds before using the T-test?
Use StatCrunch to find the T-score and the P-value. Create a pdf of your results and attach it to your post.
State a conclusion based on the context of this scenario.
StatCrunch Instructions
Download the seed data file, and then upload the file into StatCrunch.
After the data file is loaded, work through these StatCrunch directions and then use the results to answer the questions below.
Choose: Stat → T Stats → Paired.
Select Regular seed for Sample 1.
Select Kiln-dried seed for Sample 2.
For Save, check "Differences."
In Perform, select Hypothesis test for μd=μ1−μ2.
Enter a null value for μd.
For the alternative hypothesis, choose from the options of LaTeX: \ne≠, <, or > (when choosing, be sure to consider how the differences were calculated).
Press Compute!
Note that the computed differences appear in a new column called Differences.