To find the p-value for the F-value using the given ANOVA table, we need to use the F-statistic and the degrees of freedom.
Here’s a detailed, step-by-step solution for finding the p-value:
1. Identify Given Values:
- Degrees of freedom within groups ([tex]\(df_{within}\)[/tex]): 42
- Degrees of freedom between groups ([tex]\(df_{between}\)[/tex]): 2
- F-value within groups ([tex]\(F\)[/tex]): 0.416
2. Understand the Relationship:
- The F-value is used to compare the variance between the groups to the variance within the groups.
- The p-value tells us the probability of observing an F-value as extreme as, or more extreme than, the one observed, under the assumption that the null hypothesis is true.
3. Calculate the p-value:
- The p-value can be found using the F-distribution, specifically the survival function (which is [tex]\(1 - \)[/tex] cumulative distribution function) for the F-distribution.
- In R, we use the function `pf` to find the cumulative probability, and then subtract it from 1 to get the p-value.
4. Using the F-distribution:
- The p-value can be obtained using the F-distribution survival function with the given F-value and degrees of freedom.
- This is executed as `pf(F, df_between, df_within, lower.tail=FALSE)` in R.
5. Result Interpretation:
- Given the provided values, we can interpret the p-value directly.
Based on the information given, after performing the necessary calculations, the p-value is found to be approximately 0.6623684953650502.
6. Round the p-value:
- Finally, we round the p-value to the nearest thousandth.
- The p-value rounded to the nearest thousandth is 0.662.
Thus, the p-value for the F-value is 0.662, when rounded to the nearest thousandth.
