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. \(\begin{array}{lrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } \\ \text { Groups } & 3 & 360.0 & 120.0 & 1.60 \\ \text { Error } & 16 & 1200.0 & 75.0 & \\ \text { Total } & 19 & 1560.0 & & \end{array}\)

Short answer

There are 3 groups. The null hypothesis is that all group means are equal, while the alternative is that at least one group mean is different. The P-value cannot be determined directly from the provided table, it can be computed using an F-distribution table or software. The conclusion depends on the P-value calculation: if it's less than or equal to 0.05, the null hypothesis is rejected; if it's more than 0.05, the null hypothesis is not rejected.

Step-by-step solution

Identify the number of groups

The number of groups is identified from the 'DF' (Degrees of Freedom) under 'Groups'. In this case, there are 3 groups.

Formulate the hypotheses

The null hypothesis \(H_0\) states that all population means are equal, or the group variable has no effect. The alternative hypothesis \(H_1\) states that at least one population mean is different.

Compute or Identify the P-value

The P-value cannot be found directly from the given table. However, it can be calculated with software or looked up in an F-distribution table using the F value of 1.60 from the 'Groups' row, and degrees of freedom (3 for groups and 16 for error).

Test the Hypothesis and Conclude

The P-value is compared with the given level of significance (0.05). If the P-value is less than or equal to 0.05, the null hypothesis is rejected in favor of the alternative. If the P-value is more than 0.05, the null hypothesis is not rejected. We cannot explicitly do this step as the P-value is not given and needs software or a table to compute.

Key concepts

Null and Alternative Hypotheses

Understanding the null and alternative hypotheses is fundamental in hypothesis testing, including Analysis of Variance (ANOVA). Essentially, the null hypothesis (\( H_0 \)) asserts that there is no effect or difference, and any observed variation is due to chance. In ANOVA, it posits that all group means are equal. Conversely, the alternative hypothesis (\( H_1 \) or \( H_a \)) suggests that an effect exists or there is a difference; specifically, at least one group mean is different from the others.

In our case, for the comparison of means across multiple groups, the null hypothesis states that the means of all three groups are the same (\( H_0: \mu_1 = \mu_2 = \mu_3 \) where \( \mu \) represents a group mean). The alternative hypothesis contends that at least one group mean is different (\( H_1: \text{not all } \mu_i \text{ are equal} \)). The decision to reject or not to reject the null hypothesis is made based on the P-value, which we will discuss in a subsequent section.

Degrees of Freedom

Degrees of freedom (DF) are crucial for understanding the distributions used in hypothesis testing. They are essentially the number of independent values that can vary in an analysis without violating given restrictions. In the context of ANOVA, DF can be split into two parts: within groups and between groups (also known as error and groups, respectively).

The degrees of freedom associated with the groups are one less than the number of groups (DF = number of groups - 1), because once you know the means of all but one group, the last one is constrained by the overall mean. Similarly, the degrees of freedom for the error or within groups are calculated by subtracting the number of groups from the total number of observations (DF = total n - number of groups). In our exercise, the DF for groups is 3, suggesting four groups are being compared. The error DF is 16, implying that there are 20 measurements in total across the groups.

P-value

The P-value is a pivotal piece of information used in statistical hypothesis testing. It tells us the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is correct. A small P-value, typically less than or equal to 0.05, indicates strong evidence against the null hypothesis, prompting researchers to reject it in favor of the alternative hypothesis.

In the given ANOVA example, the P-value is not provided directly. It would be calculated using statistical software or an F-distribution table, considering the F statistic of 1.60 and the corresponding degrees of freedom for both groups and error. If this calculated P-value is 0.05 or lower, we conclude that the group means are significantly different at a 5% significance level. If the P-value is higher than 0.05, we would fail to reject the null hypothesis, indicating no significant difference in the group means.

Statistical Significance

Statistical significance is a term used to decide if a result is not likely to be due to random chance. This determination is typically based on a pre-defined significance level, with 0.05 being a common threshold. If our test's P-value is less than or equal to the significance level, we conclude that our results are statistically significant.

In an ANOVA context, achieving statistical significance would mean we have enough evidence to support the claim that not all group means are equal. Conversely, if we do not achieve statistical significance, we do not have sufficient evidence to reject the null hypothesis, which claims that all group means are equal. In the exercise scenario, a conclusion is drawn using a 5% significance level. We compare the P-value with 0.05 to make this decision. In the absence of an explicit P-value, an assumption must be made based on common practice for this threshold.