Hypothesis Testing for One-Way ANOVA

One-Way ANOVA Hypothesis Test

• The ANOVA table or the raw data (if you can use software)
• The number of groups or samples $k$
• The significance level $α$
• If it is not provided, you should assume it's 0.05 or see how small or large the p-value is.

Wize Tip

You must know this table:

Hypothesis Test Five Steps

• Step 1: Locate the relevant variables and identify the situation
• Determine if the question has more than two SRS, the populations are normally distributed, independent, and the variance of the populations can be assumed to be equal.
• Find the significance level $α$.
• Determine the number of samples $k$.
• Fill in or construct the ANOVA table.

• Step 2: State the hypotheses
• In this step we need to determine the null hypothesis $H_o$ and the alternative hypothesis $H_a$ (also sometimes denoted $H_1$).
• The null hypothesis
• $H_o:µ_1=\ µ_2=...=\ µ_k$ (the population means are all equal)
• This implies the treatments had no impact on the means.
• The alternative hypothesis
• $H_a:$ at least one of the means $\mu_i$ is different from the others
• This implies treatments don't all have the same impact.
• Thus, the hypothesis statement takes the form:
• $H_0:\ µ_1=µ_2=...=µ_k$
• $H_a:$ at least one of the means $\mu_i$ is different from the others

• Step 3: Calculate the Test Statistic
• Test statistic $\displaystyle F=\frac{MSB}{MSW}$
• The F-distribution has two degrees of freedom
• $df_{numerator}=df_{B}=k-1$ (degrees of freedom for "Between Groups")
• $df_{denominator} =df_{W}= n-k$ (degrees of freedom for "Within Groups")
• where $n$ is the overall sample size.

• Step 4: Decision Rule
• Find the p-value to compare with the significance level $\alpha$
• The p-value should either be given in the ANOVA table or obtained using software
• Alternatively, you can find the critical value to compare with the F statistic
• You can find the critical value using the appropriate F-table, given the significance level $\alpha$ and degrees of freedoms ($df_{numerator}, df_{denominator}$)

• Step 5: Conclusion
• If p-value$<α \rightarrow$ Reject $H_o$
• We have enough statistical evidence to conclude that at least one of the population mean differs.
• If p-value$>α \rightarrow$ Fail to reject $H_o$
• We do not have enough statistical evidence to conclude that at least one of the population mean differs.