Question

Exercises 8.46 to 8.52 refer to the data with analysis shown in the following computer output: \(\begin{array}{lrrrr}\text { Level } & \text { N } & \text { Mean } & \text { StDev } & \\ \text { A } & 6 & 86.833 & 5.231 & \\ \text { B } & 6 & 76.167 & 6.555 & \\ \text { C } & 6 & 80.000 & 9.230 & \\ \text { D } & 6 & 69.333 & 6.154 & \\ \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\ \text { Groups } & 3 & 962.8 & 320.9 & 6.64 & 0.003 \\ \text { Error } & 20 & 967.0 & 48.3 & & \\ \text { Total } & 23 & 1929.8 & & & \end{array}\) Is there evidence for a difference in the population means of the four groups? Justify your answer using specific value(s) from the output.

Short answer

Yes, there is evidence for a difference in the population means of the four groups. This is because the P-value (0.003) is lower than the typical significance level (0.05), suggesting that at least one of the population means significantly differs from the others.

Step-by-step solution

Identify the P-value

From the provided table, identify the P-value given in the row labelled 'Groups'. This value is provided in the column labelled 'P' for the variable 'Groups'. In this case, the P-value is 0.003.

Compare the P-value to the significance level

The common significance level is 0.05. Compare the computed P-value (0.003) with this significance level. In this case, 0.003 < 0.05.

Decision Rule

Since the P-value is less than the significance level (α=0.05), it suggests that at least one of the population means differs significantly from the rest. Hence, there is evidence to reject the null hypothesis. Therefore, there is evidence to conclude that the population means of the different groups are not all the same.

Key concepts

Population Means Comparison

Comparing population means is a fundamental aspect of ANOVA (Analysis of Variance), a statistical method used to analyze the differences between group means and their associated procedures.

In the context of the given exercise, the goal of ANOVA is to determine whether there are statistically significant differences among the mean scores of the four different groups (A, B, C, and D). The table from the exercise provides the means and the sample size for each group, along with the ANOVA summary output, which includes the degrees of freedom (DF), sum of squares (SS), mean squares (MS), F-statistic (F), and P-value.

The F-statistic is calculated by dividing the variance between the groups (MS of Groups) by the variance within the groups (MS of Error). The outcome of this calculation helps determine if the variance observed between group means is larger than what could be expected due to random chance.

If the calculated F-statistic is sufficiently large, it could indicate a significant difference in population means. However, to establish whether this difference is statistically significant, we must assess the P-value that corresponds with the observed F-statistic.

P-value Interpretation

The P-value is a critical concept in statistics used to interpret the results of hypothesis testing. It represents the probability of obtaining the observed results, or more extreme results, given that the null hypothesis is true.

With regard to our ANOVA analysis, the P-value is used to determine the evidence against the null hypothesis, which typically posits that there is no difference between group means. In simple terms, a smaller P-value suggests that there is stronger evidence to reject the null hypothesis.

In the provided exercise, a P-value of 0.003 is observed, which indicates a very low probability that the differences between the group means have occurred by chance alone. This P-value is crucial as it quantifies the evidence and helps decide whether the null hypothesis can be rejected. For the interpretation, the smaller the P-value, the greater the statistical significance of the observed differences.

Significance Level

The significance level, often denoted by \( \alpha \), is a threshold chosen by researchers to determine the point at which the results of a hypothesis test can be considered statistically significant. It is a pre-determined probability for erroneously rejecting the null hypothesis, known as a Type I error.

Commonly, a significance level of 0.05 (5%) is used in many scientific studies. If the P-value is less than or equal to the significance level, it suggests that the observed data are sufficiently inconsistent with the null hypothesis, and it is therefore rejected.

In the exercise, the P-value (0.003) is compared with the standard significance level (0.05). Since 0.003 is less than 0.05, this leads to the conclusion that there is statistically significant evidence against the null hypothesis. Therefore, we infer that the differences between the population means of the groups are unlikely to have arisen by random chance, signifying a meaningful difference in the means.