Question

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 depression in a group of bullies and a group of bystanders using an 18-item, 5-point depression scale. Assume scores on the depression scale are normally distributed and that the variances of the depression scores are the same among bullies and bystanders. The group of 23 bullies scored an average of 40.1 with a sample standard deviation of 10 on the depression 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 bullies or bystanders will be more depressed, so you formulate the null and alternative hypotheses as: H0: μbullies - μbystanders = 0 H1: μbullies - μbystanders ≠ 0 You conduct an independent-measures t test. Given your null and alternative hypotheses, this is a two-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 is 49. The critical t-scores that form the boundaries of the rejection region for α = 0.05 are ± 2.009. 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 s^2p = ((n1-1)s1^2 + (n2-1)s2^2) / (n1 + n2 - 2). The standard error is √(s^2p/n1 + s^2p/n2). The t statistic is (M1 - M2) / standard error. The t statistic is -2.34. The t statistic is in the rejection region. Therefore, the null hypothesis is rejected. You can conclude that bullies have a different mean depression score than bystanders. Thus, it can be said that these two means are significantly different from one another.

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 depression in a group of bullies and a group of bystanders using an 18-item, 5-point depression scale. Assume scores on the depression scale are normally distributed and that the variances of the depression scores are the same among bullies and bystanders.

The group of 23 bullies scored an average of 40.1 with a sample standard deviation of 10 on the depression 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 bullies or bystanders will be more depressed, so you formulate the null and alternative hypotheses as:

H0: μbullies - μbystanders = 0
H1: μbullies - μbystanders ≠ 0

You conduct an independent-measures t test. Given your null and alternative hypotheses, this is a two-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 is 49.

The critical t-scores that form the boundaries of the rejection region for α = 0.05 are ± 2.009.

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 s^2p = ((n1-1)s1^2 + (n2-1)s2^2) / (n1 + n2 - 2). The standard error is √(s^2p/n1 + s^2p/n2).

The t statistic is (M1 - M2) / standard error.

The t statistic is -2.34.

The t statistic is in the rejection region.

Therefore, the null hypothesis is rejected. You can conclude that bullies have a different mean depression score than bystanders. Thus, it can be said that these two means are significantly different from one another.

Added by Mark W.

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