HYPOTHESIS TESTING WITH *z* TESTS: WHO GETS BULLIED?

Who Gets Bullied?

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Welcome

Who Gets Bullied?

Authors:

Kelly M. Goedert, Seton Hall University

Susan A. Nolan, Seton Hall University

Kaylise D. Algrim, Seton Hall University

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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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).

Correct! The null hypothesis is that there is no mean difference; because this is a two-tailed test, the alternative hypothesis simply states that there is a difference, but does not indicate the direction of the difference.

Actually, the null hypothesis says that there is no mean difference; because this is a two-tailed test, the alternative hypothesis simply states that there is a difference, but does not indicate the direction of the difference.

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).

Correct! A z test involves locating a sample mean within a distribution of sample means, with the population mean at its center. In this case, the sample mean was 19.15 standard errors of the mean higher than the population mean.

Actually, a z test involves locating a sample mean within a distribution of sample means, with the population mean at its center. In this case, the sample mean was 19.15 standard errors of the mean higher than the population mean.

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).

Correct! A z test involves locating a sample mean within a distribution of sample means, with the population mean at its center. In this case, the negative sign on the z test indicates the sample mean was 17.82 standard errors of the mean lower than the population mean.

Actually, a z test involves locating a sample mean within a distribution of sample means, with the population mean at its center. In this case, the negative sign on the z test indicates the sample mean was 17.82 standard errors of the mean lower than the population mean.

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.

Correct! To calculate the standard error of the mean, we divide the population standard deviation by the square root of the sample size, the result of which is 1.011.

Actually, it’s 1.011. To calculate the standard error of the mean, we divide the population standard deviation by the square root of the sample size.

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.

Correct! The central limit theorem tells us that the shape of the distribution of sample means approaches normality if the sample size is 30 or greater. So, it is okay to perform and interpret the z test in this case.

Actually, the central limit theorem tells us that the shape of the distribution of sample means approaches normality if the sample size is 30 or greater. So, it is okay to perform and interpret the z test in this case.

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.”

Correct! Because the p value is less than .05, the typical cutoff for determining significance, we can say the results are statistically significant; therefore, we know that the researchers rejected the null hypothesis. Because we are given the means, we know the AS sample had average anxiety scores higher than those for the population.

Actually, because the p value is less than .05, the typical cutoff for determining significance, we can say the results are statistically significant; therefore we know that the researchers rejected the null hypothesis. Because we are given the means, we know the AS sample had average anxiety scores higher than those for the population.

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.

Correct! The statement of the results indicates that social vulnerability seems to predict whether a child is bullied, rather than social skills.

Actually, the statement of the results indicates that social vulnerability seems to predict whether a child is bullied, rather than social skills.

The bottom line: Although children with Asperger syndrome do have poorer social skills, on average, increased social vulnerability—and not poorer social skills—seems to be directly linked to increased bullying.

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Licensed Material is being used for illustrative purposes only; any person depicted in the licensed Material is a model.

REFERENCES

Sofronoff, K., Dark, E., & Stone, V. (2011). Social vulnerability and bullying in children with Asperger syndrome.
*Autism: The International Journal of Research and Practice, 15, *355–372.
https://doi.org/10.1177/1362361310365070