An analysis of variance is used to evaluate the mean differences for a research study
comparing three treatments with a separate sample of n = 6 in each treatment. If the data
produce an F-ratio of F = 4.10, then which of the following is the correct statistical decision?
a. Reject the null hypothesis with α = .05 but not with α = .01.
b. Reject the null hypothesis with either α = .05 or α = .01.
c. Fail to reject the null hypothesis with either α = .05 or α = .01.
d. There is not enough information to make a statistical decision.
The critical value of F1 = F (0.05, 2, 15) = 3.68 and F2 =F (0.01, 2, 15) = 6.35.
As F = 4.10 > F1 but < F2, Reject the null hypothesis with á = .05 but not with á = .01.
(a) is the correct choice.
To determine the correct statistical decision, we need to compare the obtained F-ratio with the critical F-value at a given alpha level (α).
Given that F = 4.10, we need to compare it with the critical F-values for α = 0.05 and α = 0.01.
The degrees of freedom for the numerator equal the number of treatment groups minus one (k - 1). In this case, we have 3 treatments, so the numerator degrees of freedom (df₁) would be 3 - 1 = 2.
The degrees of freedom for the denominator equal the total sample size minus the number of treatment groups (N - k). In this case, since we have 6 participants in each treatment group and 3 treatment groups, the denominator degrees of freedom (df₂) would be 6 * 3 - 3 = 15.
Looking up the critical F-values in a table or using software, the critical F-value at α = 0.05 for df₁ = 2 and df₂ = 15 is approximately 3.68. The critical F-value at α = 0.01 for the same degrees of freedom is approximately 5.79.
Since the obtained F-ratio of 4.10 is less than the critical F-value at α = 0.05 (3.68), but greater than the critical F-value at α = 0.01 (5.79), the correct statistical decision is:
a. Reject the null hypothesis with α = .05 but not with α = .01.
To determine the correct statistical decision, we need to compare the calculated F-ratio to the critical F-value at the specified significance level (alpha). In this case, the alpha levels given are α = 0.05 and α = 0.01.
The critical F-value depends on the degrees of freedom for the numerator (dfn) and the degrees of freedom for the denominator (dfd). The dfn is equal to the number of treatments minus one, which is 3 - 1 = 2. The dfd is equal to the total sample size minus the number of treatments, which is 6 * 3 - 3 = 15.
To find the critical F-value, we can use a statistical table or calculator. For both alpha levels, the critical F-values for dfn = 2 and dfd = 15 are:
α = 0.05: F_critical = 3.68
α = 0.01: F_critical = 5.83
Now, we can compare the calculated F-ratio (F = 4.10) to the critical F-values:
a. Reject the null hypothesis with α = 0.05 but not with α = 0.01.
The calculated F-ratio (4.10) is greater than the critical F-value for α = 0.05 (3.68). Therefore, we reject the null hypothesis at α = 0.05. However, we cannot make a decision for α = 0.01 until we compare it to the critical F-value.
b. Reject the null hypothesis with either α = 0.05 or α = 0.01.
The data provided in option b is incorrect as we have already determined that we can only reject the null hypothesis at α = 0.05, not at α = 0.01.
c. Fail to reject the null hypothesis with either α = 0.05 or α = 0.01.
Since we reject the null hypothesis at α = 0.05, option c is incorrect.
d. There is not enough information to make a statistical decision.
We do have sufficient information to make a statistical decision based on the given F-ratio and the alpha levels. Therefore, option d is incorrect.
Based on the analysis, the correct statistical decision is a. Reject the null hypothesis with α = 0.05 but not with α = 0.01.