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

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? $H_{0}: \mu_{1}=\mu_{2}$ $H_{0}: \mu_{1}=\mu_{2}=\mu_{3}$ $H_{0}: \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}$ $H_{A}: \mu_{1}=\mu_{2}=\mu_{3}$
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Solution

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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=1kni(XˉiXˉ)2=91.2 SS_{between} = \sum_{i=1}^k n_i (\bar{X}_i - \bar{X})^2 = 91.2

  • The sum of squares within groups is calculated as: SSwithin=i=1kj=1ni(XijXˉi)2=53.2 SS_{within} = \sum_{i=1}^k \sum_{j=1}^{n_i} (X_{ij} - \bar{X}_i)^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=SSbetweendfbetween=91.22=45.6 MS_{between} = \frac{SS_{between}}{df_{between}} = \frac{91.2}{2} = 45.6

  • The mean square within groups is given by: MSwithin=SSwithindfwithin=53.2124.4333 MS_{within} = \frac{SS_{within}}{df_{within}} = \frac{53.2}{12} \approx 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=MSbetweenMSwithin=45.64.433310.2857 F = \frac{MS_{between}}{MS_{within}} = \frac{45.6}{4.4333} \approx 10.2857

Step 4: Calculate P-value

The p-value is calculated based on the F-statistic and the degrees of freedom: P=1F(Fobserved;dfbetween,dfwithin)=1F(10.2857;2,12)0.0025 P = 1 - F(F_{observed}; df_{between}, df_{within}) = 1 - F(10.2857; 2, 12) \approx 0.0025

Step 5: Conclusion

Based on the calculated p-value, we compare it to the significance level (α=0.05\alpha = 0.05):

  • Since P<αP < \alpha, 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 \boxed{H_{0}: \mu_{1}=\mu_{2}=\mu_{3}}

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