“Bullying,” according to noted expert Dan Olweus, “poisons the educational environment and affects the learning of every child.” Bullying and victimization are evident as early as preschool, with the problem peaking in middle school. Suppose you are interested in the emotional well-being of not only the victims but also bystanders, bullies, and those who bully but who are also victims (bully-victims). You decide to measure anxiety in a group of bully-victims and a group of bystanders using an 18-item, 5-point anxiety scale. Assume scores on the anxiety scale are normally distributed and that the variances of the anxiety scores are the same among bully-victims and bystanders.
The group of 23 bully-victims scored an average of 40.1 with a sample standard deviation of 10 on the anxiety scale. The group of 28 bystanders scored an average of 46.8 with a sample standard deviation of 11 on the same scale. You do not have any presupposed assumptions about whether bully-victims or bystanders will be more anxious, so you formulate the null and alternative hypotheses as:
H0
0
: μbully-victims
bully-victims
– μbystanders
bystanders
= 0
H1
1
: μbully-victims
bully-victims
– μbystanders
bystanders
≠ 0
You conduct an independent-measures t test. Given your null and alternative hypotheses, this is atwo-tailed test. To use the Distributions tool to find the rejection region, you first need to set the degrees of freedom. The degrees of freedom is49 .
t Distribution
Degrees of Freedom = 58


-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
t
The critical t-scores that form the boundaries of the rejection region for α = 0.05 are ± .
In order to calculate the t statistic, you first need to calculate the standard error under the assumption that the null hypothesis is true. In order to calculate the standard error, you first need to calculate the pooled variance. The pooled variance is s2p
p
2
= . The standard error is s(M1 – M2)
(M1 – M2)
= .
The t statistic is .
The t statistic in the rejection region. Therefore, the null hypothesis is . You conclude that bully-victims have a different mean anxiety score than bystanders. Thus, it can be said that these two means are different from one another.