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. \(\begin{array}{lrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } \\ \text { Groups } & 2 & 540.0 & 270.0 & 8.60 \\ \text { Error } & 27 & 847.8 & 31.4 & \\ \text { Total } & 29 & 1387.8 & & \end{array}\)

Short answer

The number of groups is 2. The null hypothesis is that all group means are equal and the alternative hypothesis is that at least one group mean differs. The p-value is not provided in the table, so can't be determined. The conclusion of the test based on a 5% significance level isn't available due to absence of the p-value.

Step-by-step solution

Identify the Number of Groups

Looking at the provided output table for the ANOVA test, there is a row labeled 'Groups' with a value '2' under the 'DF' (or Degree of Freedom) column. This indicates that there are 2 groups or treatments.

State the Null and Alternative Hypotheses

In an ANOVA test, the null hypothesis states that all group means are equal, while the alternative hypothesis posits that at least one group mean is different. So, \(H_0: \mu_1 = \mu_2\) and \(H_a: \mu_1 \neq \mu_2\) (where \( \mu_1\) and \( \mu_2\) are the means of the first and second group respectively).

Determine the p-value

The ANOVA test table does not provide the p-value directly; it is typically calculated using the F statistic and the degrees of freedom for both the groups and error (except for some specific software outputs where p-value could be included). Since the p-value isn't mentioned here, we can not provide it directly. In general, software like R and Excel can be used to compute the p-value.

Conclude the Test

Based on a 5% significance level, if the determined p-value is less than 0.05, we would reject the null hypothesis, indicating there's evidence that at least one group mean is different. However, since the p-value isn't provided, the conclusion on this is not available.

Key concepts

Degrees of Freedom

Understanding degrees of freedom (DF) is crucial when working with statistical tests like ANOVA (Analysis of Variance). Degrees of freedom refer to the number of values in a calculation that are free to vary. Put simply, they're the number of 'pieces' of information that go into the estimate of a parameter.

For an ANOVA test, degrees of freedom are partitioned into two parts: one for the groups and one for the error or within groups. In the case of our exercise, with 'Groups' DF being 2, this implies that there were 3 groups being compared (since DF is one less than the number of groups). The 'Error' DF, which in this case is 27, represents the total number of observations minus the number of groups.

Why does this matter? The degrees of freedom affect the shape of the F-distribution used to calculate the p-value, and thus influence the outcome of an ANOVA test.

Null and Alternative Hypotheses

In any hypothesis test, including ANOVA, two competing hypotheses are formulated: the null hypothesis (denoted as H₀) and the alternative hypothesis (denoted as H_a).

The null hypothesis is a statement of no effect or no difference, and it's what we assume to be true before collecting any data. For an ANOVA test, the null hypothesis states that all group means are equal. In mathematical terms, it's expressed as H₀: µ₁ = µ₂ = ... = µ_k, where µ represents the group mean and k is the number of groups.

The alternative hypothesis, on the other hand, posits that there's at least one significant difference among the group means. It's denoted as H_a and often expressed as H_a: Not all µ's are equal. Hypothesis testing is all about deciding whether data provides enough evidence to reject the null hypothesis in favor of the alternative hypothesis.

P-value

The p-value is a bedrock concept in statistical hypothesis testing. It represents the probability of observing the data, or something more extreme, assuming that the null hypothesis is true. A low p-value suggests that the observed data is unlikely under the null hypothesis, thus providing evidence against the null hypothesis.

In the context of our ANOVA exercise, the F-statistic of 8.60 would be used along with degrees of freedom (2 for groups, 27 for error) to calculate the p-value. Typically, this is done with statistical software. A p-value less than the chosen significance level would mean rejecting the null hypothesis, suggesting at least one group mean differs from the others.

It's important to note that the p-value is not the probability that the null hypothesis is true, nor is it the probability that the alternative hypothesis is true. It is merely an indicator of how compatible the collected data is with the null hypothesis.

Significance Level

The significance level, often denoted by the Greek letter α (alpha), is a threshold predetermined by the researcher to decide whether to reject the null hypothesis. It's a measure of how much evidence you require before acting as if the null hypothesis is false. The most common significance level used is 0.05.

In our ANOVA example, we use a significance level of 5%. If the p-value calculated from the F-statistic and degrees of freedom is less than 0.05, we conclude that there's a statistically significant difference between at least two group means, thus rejecting the null hypothesis.

However, setting a significance level is a subjective decision that balances the risk of making a Type I error (falsely rejecting the null hypothesis) against making a Type II error (not detecting a real effect). The lower the significance level, the less chance of a Type I error, but the greater the risk of failing to detect a real effect (increasing Type II error).