Questions: Suppose a researcher wants to investigate the effect of the amount of fertilizer on the height of a common houseplant. More specifically, the researcher is interested in determining if there is a difference between the mean heights of the plants receiving one of three different fertilizer levels: high, medium, and low.
Which of the following would be the correct null hypothesis?
H0: μ1=μ2
H0: μ1=μ2=μ3
H0: μ1=μ2=μ3=μ4
HA: μ1=μ2=μ3
Transcript text: Suppose a researcher wants to investigate the effect of the amount of fertilizer on the height of a common houseplant. More specifically, the researcher is interested in determining if there is a difference between the mean heights of the plants receiving one of three different fertilizer levels: high, medium, and low.
Which of the following would be the correct null hypothesis?
$H_{0}: \mu_{1}=\mu_{2}$
$H_{0}: \mu_{1}=\mu_{2}=\mu_{3}$
$H_{0}: \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}$
$H_{A}: \mu_{1}=\mu_{2}=\mu_{3}$
Solution
Solution Steps
Step 1: Calculate Sum of Squares
The total variability in the data can be partitioned into two components: the variability between groups and the variability within groups.
The sum of squares between groups is calculated as:
SSbetween=i=1∑kni(Xˉi−Xˉ)2=91.2
The sum of squares within groups is calculated as:
SSwithin=i=1∑kj=1∑ni(Xij−Xˉi)2=53.2
Step 2: Calculate Mean Squares
Next, we calculate the mean squares for both between and within groups.
The mean square between groups is given by:
MSbetween=dfbetweenSSbetween=291.2=45.6
The mean square within groups is given by:
MSwithin=dfwithinSSwithin=1253.2≈4.4333
Step 3: Calculate F-statistic
The F-statistic is calculated as the ratio of the mean square between groups to the mean square within groups:
F=MSwithinMSbetween=4.433345.6≈10.2857
Step 4: Calculate P-value
The p-value is calculated based on the F-statistic and the degrees of freedom:
P=1−F(Fobserved;dfbetween,dfwithin)=1−F(10.2857;2,12)≈0.0025
Step 5: Conclusion
Based on the calculated p-value, we compare it to the significance level (α=0.05):
Since P<α, we reject the null hypothesis.
Final Answer
There is a significant difference between the group means. The correct null hypothesis is:
H0:μ1=μ2=μ3