Please, I would the step to do this problem.
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?
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To solve this problem and determine the conclusion at the 0.05 level of significance, you need to compare the calculated F-statistic (F = 3.95) with the critical value from the F-distribution table.
Here are the steps to solve this problem:
Step 1: Set up the null and alternative hypotheses:
- Null hypothesis (H0): There is no significant difference between the means of the three independent samples.
- Alternative hypothesis (Ha): There is a significant difference between the means of the three independent samples.
Step 2: Determine the degrees of freedom:
- Degrees of freedom between groups (DFbetween): k - 1 (where k is the number of groups/samples)
- Degrees of freedom within groups (DFwithin): Total sample size (n1 + n2 + n3) - k
In this case, there are three independent samples (k = 3), so:
- DFbetween = 3 - 1 = 2
- DFwithin = (8 + 10 + 7) - 3 = 22
Step 3: Look up the critical value from the F-distribution table:
- The critical value corresponds to the desired significance level (α) and the degrees of freedom (DFbetween and DFwithin).
- In this case, the significance level is 0.05, which is commonly used.
From the F-distribution table, with DFbetween = 2 and DFwithin = 22, the critical value is approximately 3.30.
Step 4: Compare the calculated F-statistic to the critical value:
- If the calculated F-statistic is greater than the critical value, reject the null hypothesis.
- If the calculated F-statistic is less than or equal to the critical value, fail to reject the null hypothesis.
In this case, the calculated F-statistic (F = 3.95) is greater than the critical value (3.30).
Step 5: Determine the conclusion:
- Since the calculated F-statistic is greater than the critical value, we reject the null hypothesis.
- This means that there is a significant difference between the means of the three independent samples.
Now, let's move on to the second part of your question about the p-value. Unfortunately, the information provided does not allow us to directly determine the p-value. However, we can make a general statement about it.
Based on the F-distribution tables, the most accurate statement that can be made about the p-value for this test is that it is less than 0.05. This is because the calculated F-statistic is greater than the critical value at the 0.05 level of significance. However, without more precise information, we cannot provide the exact p-value.