Two Population Means A tomato farmer with a very large farm of approximately 220
ID: 3131725 • Letter: T
Question
Two Population Means
A tomato farmer with a very large farm of approximately 2200 acres had heard about a new type of rather expensive fertilizer which would supposedly significantly increase his production. The frugal farmer wanted to test the new fertilizer before committing the large investment required to fertilize a farm of his size. He therefore selected 15 parcels of land on his property and divided them each into two portions. He bought just enough of the new fertilizer to spread over one half of each parcel and then spread the old fertilizer over the other half of each parcel. His yields in pounds per tomato plant were as follows:
Parcel New Fertilizer Old Fertilizer
1 14.2 14.0
2 14.1 13.9
3 14.5 14.4
4 15.0 14.8
5 13.9 13.6
6 14.5 14.1
7 14.7 14.0
8 13.7 13.7
9 14.0 13.3
10 13.8 13.7
11 14.2 14.1
12 15.4 14.9
13 13.2 12.8
14 13.8 13.8
15 14.3 14.0
The farmer had taken statistics many years ago when in college and consequently made a couple of mistakes when testing to find if the new fertilizer was more effective: (1) He tested the data as two independent samples, and (2) He performed a two-tailed test. He decided that he was unable to conclude that there was a difference between the two fertilizers.
What if you were the fertilizer sales representative and your job was to prove the superiority of the new product to the farmer?
(1) You should start by running the same test he did in which he came to the decision that he could not conclude a difference.
(2) Perform the test as it should have been done and find if you come to a different conclusion.
3) Explain why the results were different and why your test was a stronger and more reliable test.
Explanation / Answer
The farmer had taken statistics many years ago when in college and consequently made a couple of mistakes when testing to find if the new fertilizer was more effective: (1) He tested the data as two independent samples, and (2) He performed a two-tailed test. He decided that he was unable to conclude that there was a difference between the two fertilizers.
What if you were the fertilizer sales representative and your job was to prove the superiority of the new product to the farmer?
Pooled-Variance t Test for the Difference Between Two Means
(assumes equal population variances)
Data
Hypothesized Difference
0
Level of Significance
0.05
Population 1 Sample
Sample Size
15
Sample Mean
14.22
Sample Standard Deviation
0.5493
Population 2 Sample
Sample Size
15
Sample Mean
13.94
Sample Standard Deviation
0.5275
Intermediate Calculations
Population 1 Sample Degrees of Freedom
14
Population 2 Sample Degrees of Freedom
14
Total Degrees of Freedom
28
Pooled Variance
0.2900
Standard Error
0.1966
Difference in Sample Means
0.2800
t Test Statistic
1.4239
Two-Tail Test
Lower Critical Value
-2.0484
Upper Critical Value
2.0484
p-Value
0.1655
Do not reject the null hypothesis
The 15 parcels are paired, paired sample t test is used.
One tailed test is used.
Paired t Test
Data
Hypothesized Mean Difference
0
Level of significance
0.05
Intermediate Calculations
Sample Size
15
DBar
0.2800
Degrees of Freedom
14
SD
0.2242
Standard Error
0.0579
t Test Statistic
4.8359
Upper-Tail Test
Upper Critical Value
1.7613
p-Value
0.0001
Reject the null hypothesis
The null hypothesis is rejected.
We conclude that the new fertilizers is more effective than old fertilizers.
3) Explain why the results were different and why your test was a stronger and more reliable test.
Since the 15 parcels are paired, paired sample t test is used. we expect new fertilizers is more effective than old fertilizers .One tailed (upper tail) test is used. This is a stronger and more reliable test.
Pooled-Variance t Test for the Difference Between Two Means
(assumes equal population variances)
Data
Hypothesized Difference
0
Level of Significance
0.05
Population 1 Sample
Sample Size
15
Sample Mean
14.22
Sample Standard Deviation
0.5493
Population 2 Sample
Sample Size
15
Sample Mean
13.94
Sample Standard Deviation
0.5275
Intermediate Calculations
Population 1 Sample Degrees of Freedom
14
Population 2 Sample Degrees of Freedom
14
Total Degrees of Freedom
28
Pooled Variance
0.2900
Standard Error
0.1966
Difference in Sample Means
0.2800
t Test Statistic
1.4239
Two-Tail Test
Lower Critical Value
-2.0484
Upper Critical Value
2.0484
p-Value
0.1655
Do not reject the null hypothesis
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