Question

1. Assume we have a dataset that includes 60 observations surrounding two variables of interest: (1) Soybean yields in bushels per acre (bu/acre) and (2) fertilizer treatment. Variable (1) is quantitative while variable (2) is categorical; assume that there were four different fertilizer treatments tested. Assume also that the number of observations of each fertilizer treatment was the same for each group; i.e., 15 observations of each fertilizer treatment were collected.
   1. Write out the “Generic” null hypothesis.
2. 2. Write out the “Specific” null hypothesis.
3. 3. What are the degrees of freedom for the “Between” groups? Please show your work.
4. 4. What are the degrees of freedom for the “Within” groups? Please show your work.
5. 5. What are the “Total” degrees of freedom? Please show your work.
   6. f. Assume you conducted an ANOVA test for the dataset described above and calculated a F statistic of 6.74. Using a 5% significance level, what would be your response to the null hypothesis? Please explain your answer?

Answer 1

Solution:

Part a

The generic null hypothesis is given as below:

Null hypothesis: H₀: There is no significant difference in the population means due to different treatments.

Part b

The specific null hypothesis is given as below:

Null hypothesis: H₀: There is no any significant difference in the average Soybean yields due to four different fertilizer treatments.

Part c

There are total four groups of fertilizer treatments.

So, between degrees of freedom = k – 1 = 4 – 1 = 3

Required df = 3

Part d

Each treatment have m = 15 observations.

Total number of groups = k = 4

Within degrees of freedom = k*(m – 1)

Within degrees of freedom = 4*(15 – 1) = 4*14 = 56

Required df = 56

Part e

There are total 15*4 = 60 observations.

So, total degrees of freedom = n – 1 = 60 – 1 = 59

Required df = 59

Part f

We are given

F statistic = 6.74

df1 = 3

df2 = 56

P-value = 0.000581

(by using F-table or excel)

α = 0.05

P-value < α

So, we reject the null hypothesis

There is sufficient evidence to conclude that there is a significant difference in the average Soybean yields due to four different fertilizer treatments.