Wize University Statistics Textbook > Analysis of Variance

Hypothesis Testing for One-Way ANOVA

One-Way ANOVA Hypothesis Test

To perform a hypothesis test for the difference in several means, you will need the following information:

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.

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.

Example: One-Way ANOVA Hypothesis Testing

Max Powers is a billionaire who owns three hockey teams. Each team has coaches with very different approaches:

Team A is coached by a ruthless man with anger management issues;
Team B is coached by a retired nurse and grandmother of three;
Team C is coached by a yoga instructor

To test if his three teams differ in performance, Max compared them based on their average total points per season:

\begin{array}{|l|c|c|}\hline & \textbf{Mean Points} & \textbf{Standard Deviation}\\\hline \textbf{Team A} &56&15.56\\\hline \textbf{Team B}&42&14.30\\\hline \textbf{Team C}&77&8.94\\\hline \end{array}

Step 1: Locate the relevant variables and identify the situation

The following information is provided in the question:
$α=0.05\$ (No significance level provided so we assumed this)
$k=3$
ANOVA table is provided.
In this situation we know we have to use a one-way ANOVA test because:
The question has more than two SRS.
Populations are normally distributed (we aren't told anything in particular so we can assume).
Populations are independent (again nothing specific is said so you can simply assume).
The variance of the populations can be assumed to be equal (again nothing specific is said so you can simply assume).

Step 2: State the hypotheses

$H_0:\ µ_1=µ_2=µ_3$
$H_1:\ At\ least\ one\ of\ the\ means\ is\ different\ from\ the\ others$

Step 3: Calculate the Test Statistic

$F=\displaystyle\frac{MSB}{MSW}=\displaystyle\frac{2943.33}{175.55}=16.77$
$df\ numerator\ =k-1=3-1=2$
$df\ deno\min ator\ =N-k=30-3=27$

Step 4: Decision Rule

P-Value $=0.00002$ (given in ANOVA table)

Step 5: Conclusion

$0.00002\ <\ 0.05\ =>\ pvalue\ \le\ α$ therefore we reject H0. At the 5% level of significance, we have enough evidence to reject H0. At least one of the means differs from the other ones.

*Note that since we p-value is provided in the ANOVA table we could have skipped Steps 3, 4 and used that directly for the conclusion.

Extra Practice

Hypothesis testing for one-way ANOVA

The English Premiere League and the Spanish Liga record the number of goals opportunities for several games. The league administrators want to determine whether the variance of goals opportunities in the two leagues are equal at the 5% significance level.

$n_1=11\ and\ n_2=15$
$s_1=7.21\ and\ s_2=8.32$

Hypothesis Testing for One-way ANOVA

The Real Housewives of Beverly Hills, Orange County, Atlanta, and New York are arguing in a limo about which city charges the most for nose jobs.

Which of the following can be stated as the hypotheses:

Hypothesis Testing: One-way ANOVA

The Real Housewives of Beverly Hills, Orange County, Atlanta, and New York are arguing in a limo about which city charges the most for nose jobs.

Using the results from Question 3, if $\alpha=0.01$ select the correct conclusion from the following:

Hypothesis Testing: One-way ANOVA

You wish to compare the hourly sales of 5 fast-food restaurants to see if they are different. Here is the summarized data:

Given that the p-value of the test-statistic is 0.11, the appropriate conclusion at the 10% level of significance is

Hypothesis Testing for One-way ANOVA

You wish to compare the hourly sales of 5 fast-food restaurants to see if they are different. Here is the summarized data:

Write the appropriate Ho and Ha

Hypothesis Testing: One-way ANOVA

Dr. Anova is teaching several sections of his statistics courses. His midterm averages haven't been great this year. He asks you to help him decide if the means of the classes are different.

a) How many classes does he teach?

b) How many students does he have?

Hypothesis Testing: One-way ANOVA

5 new pills are tested with the aim of reducing blood pressure.

4 participants are assigned to each pill and their blood pressure is measured.

Here is some of the data:

Hypothesis Testing for One-way ANOVA

We want to compare the typing speed of paralegals on different laptops. We sample 5 paralegals, and have them type on 4 different laptops, recording the number of words per minute they type. The results are shown below:

Conduct a Block Design test at the 5% significance level. The ANOVA results are below:

Confidence interval and Hypothesis testing

Mr. Emin loves going to Starbucks drive-thru on his busy mornings but he can’t help feeling frustrated by the amount of time he has to wait to be served. He decides to go to Starbucks for 5 days and Tim Hortons for 7 days. He records the length of time, rounded to the nearest minute he has to wait each day to be served. Here is the data

The standard deviation for Starbucks waiting time is 3.082.

The standard deviation for Tim Hortons waiting time is 2.160.

Hypothesis Testing for One-way ANOVA

A math teacher would like to know whether or not different teaching methods have an effect on his students’ marks. He randomly samples 12 students and teaches one-third of them the traditional blackboard note-taking method, one-third are instructed to learn with their peers as a group, and the remaining 4 are told to read his notes online. After one week he decides to give them a quiz to see if there are differences in the mean quiz scores.

Here are the results:

a) State the null and alternative hypothesis.

Hypothesis Testing: One-way ANOVA

A study was conducted to analyze the average decrease in blood pressure in patients with stage 2 hypertension. 5 different medications were tested, each medication being tested on 7 patients. The ANOVA F-test was used to test for any difference in mean decreases in the blood pressures across the 5 medications. What is the distribution of the test statistic under the null hypothesis?

Hypothesis Testing: One-way ANOVA

A bookstore wants to know if the mean sales are the same across 4 genres, which are action and adventure, crime, horror, and drama. Here is a limited anova table:

If the p-value is 0.048, what is your conclusion at alpha = 5% ?

Hypothesis Testing: One-way ANOVA

Which of the following is not a correct alternative hypothesis for an analysis of variance F-test?

Hypothesis Testing: One Way Anova

Here are the results:

Hypothesis Testing: One-way ANOVA

The Real Housewives of Beverly Hills, Orange County, Atlanta, and New York are arguing in a limo about which city charges the most for nose jobs.

Which of the following can be stated as the hypotheses:

Hypothesis Testing: One-way ANOVA

There are four candidates running for President of Panem. It is suspected that some candidates are favoured over others based on who got the most speaking time during the debates. The average airtime for each candidate based on the last several debates is summarized here:

Given that the sum of squares of the treatments (between) is 1 354.125 and the sum of squares of the errors (within) 298.83333,

a) What is the value of the test-statistic? Round to 3 decimal places

Hypothesis testing: one-way ANOVA

The Real Housewives of Beverly Hills, Orange County, Atlanta, and New York are arguing in a limo about which city charges the most for nose jobs.

Using a p-value of 0.04093, the appropriate conclusion at the 5% level of significance is

Hypothesis Testing: One-way ANOVA

The London Transit Commission would like to know if ridership varies by bus routes. Four routes with low ridership are sampled in 7 random days of the month.

Some of the results are listed below in the anova table:

What are the degrees of freedom of the F-statistic?

Hypothesis testing: One-way ANOVA

You wish to compare the hourly sales of 5 fast-food restaurants to see if they are different. Here is the summarized data:

Given that the p-value of the test-statistic is 0.11, the appropriate conclusion at the 10% level of significance is

Hypothesis Testing: One-way ANOVA

The London Transit Commission would like to know if ridership varies by bus routes. Four routes with low ridership are sampled in 7 random days of the month.

Some of the results are listed below in the anova table:

Compute the F-statistic.

Hypothesis Testing: One-way ANOVA

You wish to compare the hourly sales of 5 fast-food restaurants to see if they are different. Here is the summarized data:

Write the appropriate Ho and Ha

Hypothesis Testing: One Way Anova

Here are the results:

Hypothesis Testing for One-Way ANOVA

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One-Way ANOVA Hypothesis Test

Hypothesis Test Five Steps

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Example: One-Way ANOVA Hypothesis Testing

Practice: One-Way ANOVA

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Extra Practice