The mathematical version of the alternative is. We need to be careful how we set up the alternative. With more than two groups, the research question is “Are some of the means different?." If we set up the alternative to be \(\mu_1\ne\mu_2\ne…\ne\mu_t\), then we would have a test to see if ALL the means are different.
If we wanted to see if two population means are different, the alternative would be \(\mu_1\ne\mu_2\). The alternative, however, cannot be set up similarly to the two-sample case. Therefore, the null hypothesis for analysis of variance for \(t\) population means is: In one-way ANOVA, we want to compare \(t\) population means, where \(t>2\). Recall that for a test for two independent means, the null hypothesis was \(\mu_1=\mu_2\). Since such analysis is based on the analysis of variances for the data set, we call this statistical method the Analysis of Variance (or ANOVA).ġ0.2 - A Statistical Test for One-Way ANOVA 10.2 - A Statistical Test for One-Way ANOVAīefore we go into the details of the test, we need to determine the null and alternative hypotheses. Since the between-sample-variation from Lab Sloppy is large compared to the within-sample-variation for data from Lab Precise, we will be more inclined to conclude that the three population means are different using the data from Lab Precise. We need to compare the between-sample-variation to the within-sample-variation. The sample means from the two labs turned out to be the same and thus the differences in the sample means from the two labs are zero.įrom which data set can you draw more conclusive evidence that the means from the three populations are different? This example tests if the mean number of hours worked per week differs across levels of education status.Lab Sloppy also took six samples from each of the three brands and got the following measurements: Sample Due to an increased risk of Type-I errors (rejecting a true null hypothesis), when conducting multiple pairwise tests, it is recommended to use a correction, such as the Bonferroni correction, Fisher’s least significant difference (LSD), or Tukey’s procedure. Post-hoc testing can accomplish this with pairwise comparison tests (independent t-tests). The number of possible pairwise comparisons is equal to: k( k-1)/2. If an ANOVA results in a significant F-statistic, which indicates that there is some difference in means, it’s common to investigate which pairs of groups have significantly different means. If the normality and/or equal variance assumption is violated, the non-parametric Kruskal-Wallis test can be run instead of ANOVA.
H A: At least one population mean is different, or μ i ≠ μ j for some i, jĭegrees of freedom: Group: k – 1 Error: N – k. H o: The population means of all groups are equal, or μ 1 = μ 2 = … = μ k Note that we could not run a two-sample independent t-test because there are more than two groups. For example, suppose we wanted to know if the mean GPA of college students majoring in biology, chemistry, and physics differ. A one-way (or single-factor) ANOVA can be run on sample data to determine if the mean of a numeric outcome differs across two or more independent groups.
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