Chapter 1. Hypothesis Testing with z Tests: Who Gets Bullied?

1.1 Hypothesis Testing with z Tests: Who Gets Bullied?

Being a kid is often associated with innocence and playfulness, but it’s not all fun and games. Kids have the potential to be very mean to each other. Indeed, bullying is seen as a substantial problem in schools. Who is most likely to be bullied? Research has identified two types of children who tend to be bullies’ targets: (1) kids who are weaker, more insecure, more anxious, better at school, but not better at sports; and (2) kids who are more likely to be irritating, provocative, and poor at social reciprocity.

Kate Sofronoff and her fellow researchers (2011) point out that both of these sets of qualities are observed in children with Asperger syndrome (AS; a diagnosis that is now included as part of the more general autism spectrum disorder). Children with AS typically have a normal IQ, but often lack social and communication skills. The researchers wondered whether children with AS might be particularly susceptible to bullying. As a first step in answering that question, the researchers sought to empirically demonstrate that children with AS have poorer social skills, on average, than typically developing children.

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Question 1.1

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1.2

The researchers recruited a sample of 92 parents who had a school-aged child with Asperger syndrome (AS). The parents completed a series of questionnaires about their child’s behavior and social skills. For many of these questionnaires, normative data were available regarding the population mean and standard deviation. As a first step, the researchers wanted to determine whether the children with AS truly had poorer social skills and more behavior problems, on average, than typically developing children. The researchers wrote, “Single sample z tests were performed to determine whether the mean scores of the children with AS in the current sample differed significantly from the mean scores in the normative samples” (p. 364).

1.3

One of the scales that the parents completed for their children was the Spence Children’s Anxiety Scale (SCAS-P). The researchers wrote, “A single sample z test revealed that the average score on the SCAS-P for the sample of children with AS significantly differed from the mean of the normative sample, z = 19.15, p < .05” (pp. 364–365).

1.4

Another scale that the parents completed for their children was the Spence Social Skills Questionnaire (SSSQ-P). The researchers wrote, “A single sample z test showed the scores on the SSSQ-P for the children with AS were significantly different from the mean of the normative population, z = –17.82, p < .05” (p. 365).

1.5

In a table, the researchers reported the mean and the standard deviation of the anxiety scale for the sample (M = 33.56, SD = 19.40) as well as for the population (µ = 14.20, σ = 9.70). Recall that the sample size is N = 92.

1.6

Recall that the researchers reported the mean and the standard deviation of the anxiety scale for the sample (M = 33.56, SD = 19.40) as well as for the population (µ = 14.20, σ = 9.70), and that the sample size was N = 92. As mentioned on a previous screen, the researchers reported, “A single sample z test revealed that the average score on the SCAS-P for the sample of children with AS significantly differed from the mean of the normative sample, z = 19.15, p < .05” (pp. 364–365). One assumption when performing a z test is that scores in the underlying population from which the sample is drawn are approximately normally distributed. These researchers did not report any tests of this assumption. Let’s assume the worst-case scenario—that the anxiety scores in the population are not normally distributed.

1.7

Here are the statistics for the anxiety scale again, with means and standard deviations for the sample (M = 33.56, SD = 19.40) and for the population (µ = 14.20, σ = 9.70). With their sample size of N = 92, the researchers reported the following results from a z test: “z = 19.15, p < .05.”

1.8

To look at the relationship between social skills and bullying, the researchers performed some additional statistical tests. They found that “only social vulnerability significantly predicted bullying on its own, . . . p < .001. Anger, anxiety, social skills, and intensity and number of behavior problems all failed to independently predict bullying” (pp. 366–367). The researchers’ original research question was whether poorer social skills would lead to an increased likelihood that a child would be bullied.

1.9

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