To find the value of the test statistic for comparing the means of the three groups (Direct, Indirect, and Combination), we will perform a one-way ANOVA (Analysis of Variance). ANOVA tests whether the means of three or more groups are significantly different from each other. Here's the process:

Step 1: State the hypothesis

• Null hypothesis $H_0$: All group means are equal ($\mu_1 = \mu_2 = \mu_3$).
• Alternative hypothesis $H_a$: At least one group mean is different.

Step 2: Organize the data

• Direct: 17.0, 16.5, 25.1, 15.8, 18.3, 20.2, 15.9, 14.1
• Indirect: 18.4, 22.2, 24.0, 20.4, 21.4, 18.3, 24.3, 19.7, 20.7
• Combination: 25.1, 24.0, 26.9, 27.4, 25.7, 24.5

Step 3: Calculate ANOVA

We compute the following:

1. Grand Mean ($\bar{x}_\text{grand}$): Mean of all observations.
2. Between-group variance (SSB): Variability due to differences between group means.
3. Within-group variance (SSW): Variability within each group.
4. Degrees of freedom:
   • Between groups: $df_B = k - 1$ where $k$ is the number of groups.
   • Within groups: $df_W = N - k$ where $N$ is the total number of observations.
5. Mean square values:
   • $MS_B = \frac{SSB}{df_B}$
   • $MS_W = \frac{SSW}{df_W}$
6. F-statistic: $F = \frac{MS_B}{MS_W}$

Let's calculate these values. numpy as np from scipy.stats import f_oneway

Data

direct = [17.0, 16.5, 25.1, 15.8, 18.3, 20.2, 15.9, 14.1] indirect = [18.4, 22.2, 24.0, 20.4, 21.4, 18.3, 24.3, 19.7, 20.7] combination = [25.1, 24.0, 26.9, 27.4, 25.7, 24.5]

Perform ANOVA

f_statistic, p_value = f_oneway(direct, indirect, combination)

Round the F-statistic to two decimal places

f_statistic_rounded = round(f_statistic, 2) f_statistic_roundedThe value of the test statistic (F-statistic) is 15.86.

Would you like a detailed breakdown of the ANOVA calculations or any other related information?

Here are 5 follow-up questions you might explore:

1. What does the F-statistic value tell us about the group means?
2. How do we interpret the p-value associated with this F-statistic?
3. What are the assumptions underlying the ANOVA test?
4. How would the analysis change if we used a non-parametric test?
5. How can post-hoc tests help in identifying which groups differ?

Tip: Always check if the assumptions for ANOVA (normality and homogeneity of variance) are met before interpreting the results.