Question

self - esteem for adolescents (mcgee, williams, howden - chapman, martin, & kawachi, 2006). in a representative study, a sample of $n = 100$ adolescents with a history of group participation is given a standardized self - esteem questionnaire. for the general population of adolescents, scores on this questionnaire form a normal distribution with a mean of $mu = 50$ and a standard deviation of $sigma = 15$. the sample of group - participation adolescents had an average of $m = 53.8$.
a. does this sample provide enough evidence to conclude that self - esteem scores for these adolescents are significantly different from those of the general population? use a two - tailed test with $alpha =.05$.
b. compute cohen’s $d$ to measure the size of the difference.
c. write a sentence describing the outcome of the hypothesis test and the measure of effect size as it would appear in a research report.

Explanation:

Step1: State the hypotheses

The null hypothesis \(H_{0}:\mu = 50\) and the alternative hypothesis \(H_{1}:\mu
eq50\)

Step2: Calculate the standard error

The standard error \(SE=\frac{\sigma}{\sqrt{n}}\), where \(\sigma = 15\) and \(n = 100\). So \(SE=\frac{15}{\sqrt{100}}=\frac{15}{10} = 1.5\)

Step3: Calculate the z - score

The z - score \(z=\frac{M-\mu}{SE}\), with \(M = 53.8\), \(\mu=50\) and \(SE = 1.5\). So \(z=\frac{53.8 - 50}{1.5}=\frac{3.8}{1.5}\approx2.53\)

Step4: Determine the critical z - values

For a two - tailed test with \(\alpha=0.05\), the critical z - values are \(z=\pm1.96\)

Step5: Make a decision

Since \(|z| = 2.53>1.96\), we reject the null hypothesis.

Step6: Calculate Cohen's \(d\)

Cohen's \(d=\frac{M-\mu}{\sigma}\), with \(M = 53.8\), \(\mu = 50\) and \(\sigma=15\). So \(d=\frac{53.8 - 50}{15}=\frac{3.8}{15}\approx0.25\)

Answer:

a. Yes, the sample provides enough evidence to conclude that self - esteem scores for these adolescents are significantly different from those of the general population.
b. Cohen's \(d\approx0.25\)
c. The results of the two - tailed z - test (\(z = 2.53\), \(p<0.05\)) indicate that the self - esteem scores of adolescents with a history of group participation are significantly different from the general population (\(\mu = 50\)). The effect size, measured by Cohen's \(d\), is \(d = 0.25\), suggesting a small to medium effect.