Question

Some computer output for an analysis of variance test to compare means is given. (a) How many groups are there? (b) State the null and alternative hypotheses. (c) What is the p-value? (d) Give the conclusion of the test, using a \(5 \%\) significance level. \(\begin{array}{lrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } \\ \text { Groups } & 4 & 1200.0 & 300.0 & 5.71 \\ \text { Error } & 35 & 1837.5 & 52.5 & \\ \text { Total } & 39 & 3037.5 & & \end{array}\)

Short answer

There are 4 groups. Null hypothesis: all group means are equal, alternative hypothesis: at least one group mean is different. The p-value cannot be determined from the information given. With an F-value of 5.71, there is a suggestion of possible variance among group means; however, without the p-value, a conclusive decision can't be drawn for the \(5 \% \) significance level.

Step-by-step solution

Identify Number of Groups

One can directly look at the 'DF' (Degrees of Freedom) value corresponding to 'Groups' in the table. This represents the number of groups in our analysis. In this case, it's '4' which means we have 4 groups.

Set up null and alternative hypothesis

In an Analysis of Variance test, the null hypothesis \(H_o \) claims that all group means are the same. The alternative hypothesis \(H_1 \) states that at least one group mean is different. Thus, we can write: \n- Null Hypothesis \((H_o) \): \(\mu_1 = \mu_2 = \mu_3 = \mu_4\)\n- Alternative Hypothesis \((H_1) \): at least one \(\mu_i\) is different, where \(i\) = 1, 2, 3, 4.

Calculate P-value

Unluckily, this table contains no information regarding the p-value. Typically, the p-value is calculated using the F-value and the relevant degrees of freedom. Without the proper statistical tables or software, we cannot calculate the p-value ourselves in here.

Determining Test Conclusion

Again, without the calculated p-value, it is impossible to conclusively say if we accept or reject the null hypothesis at a \(5 \% \) significance level. A general guideline is that if the F-value is larger that 1, it means that the between-group variance is higher than the within-group, suggesting could consider rejecting the null hypothesis. But it's an indication, the final conclusion is based on the comparison with the significance level. In this case, the F-value is 5.71.