10. a researcher reports an f - ratio with df = 4, 62 from an independent - measures research study.
a. how many treatment conditions were compared in the study?
b. what was the total number of participants in the study?
c. use appendix b to find the critical value for f. use α =.05.
d. what is the critical region for α =.01?

Question

Accepted Answer

### Part (a)
# Explanation:
## Step1: Recall df between groups formula
For an independent - measures ANOVA, the degrees of freedom between groups (treatment conditions) is given by $df_{between}=k - 1$, where $k$ is the number of treatment conditions.
## Step2: Solve for k
We know that $df_{between}=4$. From the formula $df_{between}=k - 1$, we can solve for $k$ by adding 1 to both sides of the equation. So $k=df_{between}+ 1$.
Substituting $df_{between} = 4$ into the formula, we get $k = 4+1=5$.
# Answer:
The number of treatment conditions is $\boldsymbol{5}$.

### Part (b)
# Explanation:
## Step1: Recall df within groups formula
For an independent - measures ANOVA, the degrees of freedom within groups is given by $df_{within}=N - k$, where $N$ is the total number of participants and $k$ is the number of treatment conditions. We know that $df_{within}=62$ and from part (a) we found that $k = 5$.
## Step2: Solve for N
We start with the formula $df_{within}=N - k$. Rearranging the formula to solve for $N$, we get $N=df_{within}+k$.
Substituting $df_{within}=62$ and $k = 5$ into the formula, we have $N=62 + 5=67$.
# Answer:
The total number of participants is $\boldsymbol{67}$.

### Part (c)
# Explanation:
## Step1: Identify df values
We have $df_{between}=4$ (numerator degrees of freedom) and $df_{within}=62$ (denominator degrees of freedom), and $\alpha = 0.05$.
## Step2: Use F - distribution table
When we look up the critical value for an F - ratio in the F - distribution table, with $df_1 = 4$ (numerator) and $df_2=62$ (denominator) and $\alpha=0.05$. Since 62 is not a common value in some tables, we can use the approximation for large $df_2$ or use the fact that as $df_2$ increases, the critical value approaches the value for $df_2=\infty$. Looking at the F - table for $\alpha = 0.05$, $df_1 = 4$ and $df_2=\infty$, the critical value is approximately 2.41. (If we use a more precise method or a table that includes $df_2 = 60$ (since 62 is close to 60), for $df_1 = 4$ and $df_2=60$, the critical value is also approximately 2.41)
# Answer:
The critical value for $F$ with $\alpha = 0.05$ is approximately $\boldsymbol{2.41}$.

### Part (d)
# Explanation:
## Step1: Identify df values
We have $df_{between}=4$ (numerator degrees of freedom) and $df_{within}=62$ (denominator degrees of freedom), and $\alpha = 0.01$.
## Step2: Use F - distribution table
We look up the critical value for an F - ratio in the F - distribution table with $df_1 = 4$ (numerator) and $df_2 = 62$ (denominator) and $\alpha=0.01$. Using the approximation for large $df_2$ (or using $df_2 = 60$ as an approximation since 62 is close to 60), for $df_1=4$ and $df_2 = 60$ and $\alpha = 0.01$, the critical value is approximately 3.29. The critical region is the set of F - ratios greater than this critical value. So the critical region is $F>3.29$ (approximate value, more precise value can be obtained using statistical software or more detailed tables).
# Answer:
The critical region for $\alpha = 0.01$ is $F>\boldsymbol{3.29}$ (approximate, depending on the table used).

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QUESTION IMAGE

Question

Explanation:

Part (a)

Step1: Recall df between groups formula

Step2: Solve for k

Step1: Recall df within groups formula

Step2: Solve for N

Step1: Identify df values

Step2: Use F - distribution table

Answer:

Part (b)