14A Preview

Preparing for the next class

In the next in-class activity, you will need to understand when to consider using a one way ANOVA and how to write the null and alternative hypotheses for a one-way  ANOVA.

In In-Class Activities 13.C and 13.D, we explored hypothesis tests that allowed us to  compare means from two groups/populations. More specifically, we performed  calculations to determine if there was evidence that the means associated with the  populations were statistically different from one another.

In the next few in-class activities, we will learn about one-way ANOVA (analysis of  variance), which is a statistical test for comparing and making inferences about means  associated with two or more groups. The one-way ANOVA is also referred to as the  one-factor ANOVA.

Question 1

1) Suppose a researcher wants to investigate the effect of the amount of fertilizer on  the height of a common houseplant. More specifically, the researcher is interested in  determining if there is a difference between the mean heights of the plants receiving  one of three different fertilizer levels: high, medium, and low.

Part A: True or False: The researcher should consider conducting a two-sample t test in this situation.

Part B: True or False: The researcher should consider conducting a one-way  ANOVA in this situation.

The null hypothesis for a one-way ANOVA states that all the group/population means are the same. This can be written as:

[latex]H_{0}:\mu_{1}=\mu_{2}=\ldots=\mu_{k}[/latex]

where [latex]k[/latex] is the number of independent groups or samples.

Question 2

2) Given the previous fertilizer scenario, which of the following would be the correct null hypothesis?

  1. a) �0: �1 = �2
  2. b) �”: �1 = �2 = �3
  3. c) �0: �1 = �2 = �3 = �4
  4. d) �+: �1 = �2 = �3

The alternative hypothesis for a one-way ANOVA is a bit different than the alternative  hypothesis we used when comparing only two group means (i.e., two-sample t-test).

When there were only two group means to consider, the null hypothesis that the two  means were the same was [latex]H_{0}:\mu_{1}=\mu_{2}[/latex]. If you wanted to show that the two means were  different or not equal, the alternative hypothesis would be [latex]H_{A}:\mu_{1}\neq\mu_{2}[/latex]. If we rejected the null hypothesis, we would be able to conclude that the two means were statistically  different.

When we reject the null hypothesis for a one-way ANOVA, we cannot simply state that  all of the means are not equal. That is, when we reject the null hypothesis, [latex]H_{0}:\mu_{1}=\mu_{2}=\ldots=\mu_{k}[/latex], we are not able to differentiate whether one of the means is  different from the others, whether two of the means are different from the others,  whether three of the means are different from the others, etc.

So, to provide flexibility and to account for the multiple outcomes associated with  rejecting the null hypothesis, the alternative hypothesis for a one-way ANOVA should be  written as:

[latex]H_{A}:[/latex] At least two of the group means are different.

Question 3

3) Given the previous fertilizer scenario, which of the following would be the correct  alternative hypothesis?

  1. a) �+: �1 ≠ �2
  2. b) �+: �1 ≠ �2 ≠ �3
  3. c) �+: All of the group means are different.
  4. d) �+: At least two of the group means are different.

Question 4

4) A researcher would like to determine whether there is a difference between the  means of exercise hours per week among people in the U.S. regions of the  Northeast, South, West, and Midwest.

Which of the following best explains why a one-way ANOVA should be considered  for this situation?

a) The researcher is interested in comparing the means of two groups.

b) The researcher is interested in comparing the means of three groups.

c) The researcher is interested in comparing the means of four groups.

Question 5

5) Which of the following is the correct null hypothesis for the scenario described in  Question 4?

  1. a) �0: �1 = �2
  2. b) �”: �1 = �2 = �3
  3. c) �0: �1 = �2 = �3 = �4
  4. d) �0: �1 ≠ �2 ≠ �3 ≠ �4

Question 6

6) Which of the following is the correct alternative hypothesis for the scenario described  in Question 4?

  1. a) �+: At least two of the group means are different.
  2. b) �+: At least three of the group means are different.
  3. c) �+: All of the group means are different.
  4. d) �+: �% ≠ �’ ≠ �. ≠ �/

Question 7

7) A researcher is interested in conducting a one-way ANOVA to compare means  between five groups. Which of the following would be the null and alternative  hypotheses for this situation?

  1. a) �”: �% = �’ = �. = �/ = �0

�+: The group means are different.

  1. b) �0: �1 = �2 = �3 = �4 = �0

�+: At least three of the group means are different.

  1. c) �”: �% = �’ = �. = �/ = �0

�+: �% ≠ �’ ≠ �. ≠ �/ ≠ �0

  1. d) �0: �1 = �2 = �3 = �4 = �0

�+: At least two of the group means are different.