In a one-way ANOVA, there are three independent samples, with n1 8, n2 = 10, and n3 = 7. The calculated F-statistic is F = 3.95. At the 0.05 level of significance, what conclusion would be reached? Based on the F distribution tables, what is the most accurate statement that can be made about the p-value for this test?
Since there are k=3 groups and N=25 observations, the F-statistic has degrees of freedom 2, 22. From the table, the p-value is 0.0343. That means that we reject the null hypothesis. The three group means are unequal.
To determine the conclusion and p-value for a one-way ANOVA, you need to compare the calculated F-statistic with the critical F-value and examine the p-value.
1. Conclusion:
To determine the conclusion, you compare the calculated F-statistic (F = 3.95) with the critical F-value at the chosen level of significance (0.05).
Using the degrees of freedom (df) for the numerator (k - 1) and denominator (N - k), where k is the number of groups and N is the total sample size:
df numerator = k - 1 = 3 - 1 = 2
df denominator = N - k = (8 + 10 + 7) - 3 = 22
Using an F distribution table or statistical software, you can find the critical F-value for df numerator = 2 and df denominator = 22 at the 0.05 level of significance.
Suppose the critical F-value is F_critical = 3.22 (just hypothetical, as the actual value is not given in the question).
Since the calculated F-statistic (F = 3.95) is greater than the critical F-value (F_critical = 3.22), the conclusion is that there is sufficient evidence to reject the null hypothesis.
2. p-Value:
The p-value is a measure of the strength of evidence against the null hypothesis. It indicates the probability of obtaining a test statistic as extreme as the observed value (in this case, F = 3.95) under the null hypothesis.
Since the exact p-value is not provided in the question, we can only make a general statement based on the F distribution table. The most accurate statement regarding the p-value is that it is less than 0.05, indicating that the result is statistically significant at the 0.05 level of significance.
However, to obtain the precise p-value, you would need more information or use statistical software that can calculate it directly.
To determine the conclusion and the p-value for the one-way ANOVA test, we need to compare the calculated F-statistic to the critical F-value and consult the F distribution table.
Here are the steps to get the conclusion and p-value:
Step 1: Determine the degrees of freedom (df) for the numerator and denominator:
- The numerator df is the number of groups minus 1: df1 = k - 1 = 3 - 1 = 2 (where k is the number of groups).
- The denominator df is the total sample size minus the number of groups: df2 = N - k = (8 + 10 + 7) - 3 = 22 - 3 = 19 (where N is the total sample size).
Step 2: Determine the critical F-value for the given alpha level (0.05) and the degrees of freedom:
- From the F distribution table, find the critical F-value for the numerator df (2) and denominator df (19) at alpha = 0.05. Let's call this critical F-value F_critical.
Step 3: Compare the calculated F-statistic (F = 3.95) to the critical F-value (F_critical):
- If the calculated F-statistic is greater than the critical F-value (F > F_critical), then we have enough evidence to reject the null hypothesis.
Step 4: Determining the conclusion:
- If the calculated F-statistic (F = 3.95) is greater than the critical F-value (F > F_critical), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 5: Determining the p-value:
- To find the p-value, we need to calculate the area under the F-distribution curve to the right of the calculated F-statistic (F = 3.95) for the degrees of freedom (2, 19).
- We can use statistical software or online calculators to find the p-value accurately.
Based on the F distribution table, we cannot determine the exact p-value for F = 3.95 because it falls between two tabulated values. The most accurate statement we can make is that the p-value for this test is between the p-values associated with the two tabulated F-values. The exact p-value can be obtained using statistical software or online calculators.