Childhood participation in sports, cultural groups,

and youth groups appears to be related to improved
self-esteem for adolescents (McGee, Williams,
Howden-Chapman, Martin, & Kawachi, 2006). In
a representative study, a sample of n 5 100 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 m 5 50 and a standard deviation of s 5 15. The
sample of group-participation adolescents had an
average of M 5 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 a 5 .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.

It would help if you proofread your questions before you posted them.

I assume you are using "5" instead of "=".

Z = (mean1 - mean2)/standard error (SE) of difference between means

SEdiff = √(SEmean1^2 + SEmean2^2)

SEm = SD/√n

If only one SD is provided, you can use just that to determine SEdiff.

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.

a. To determine if the sample of group-participation adolescents has significantly different self-esteem scores compared to the general population, we can conduct a two-tailed t-test.

Step 1: State the hypotheses:
- Null hypothesis (H₀): The self-esteem scores of the group-participation adolescents are not significantly different from those of the general population (μ = 50).
- Alternative hypothesis (H₁): The self-esteem scores of the group-participation adolescents are significantly different from those of the general population (μ ≠ 50).

Step 2: Set the significance level (α) to .05.

Step 3: Calculate the test statistic:
We can calculate the test statistic (t-value) using the formula:
t = (M - μ) / (s / √n)
where M = sample mean, μ = population mean, s = population standard deviation, and n = sample size.

For the given sample: M = 53.8, μ = 50, s = 15, n = 100.

t = (53.8 - 50) / (15 / √100)
t = 3.8 / 1.5
t ≈ 2.53

Step 4: Determine the critical value:
Since it is a two-tailed test, with a significance level of .05, we divide α by 2, resulting in .025. We can then find the critical value corresponding to .025 in the t-distribution table, considering 99 degrees of freedom (n - 1).

Based on the table, the critical value (t_crit) is approximately ±1.984.

Step 5: Make a decision:
Compare the absolute value of the test statistic to the critical value. If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis. If it is less than the critical value, we fail to reject the null hypothesis.

In this case, |2.53| > 1.984, so we reject the null hypothesis.

Therefore, there is enough evidence to conclude that self-esteem scores for these adolescents are significantly different from those of the general population.

b. Cohen's d is used to measure the effect size, which quantifies the magnitude of the difference between means.
To compute Cohen's d, we can use the formula:
d = (M - μ) / s
where M = sample mean, μ = population mean, and s = population standard deviation.

For the given sample: M = 53.8, μ = 50, and s = 15.

d = (53.8 - 50) / 15
d ≈ 0.22

Therefore, the Cohen's d value for the difference in self-esteem scores is approximately 0.22.

c. A possible sentence describing the outcome of the hypothesis test and the measure of effect size for a research report could be:
"The results of the hypothesis test indicated that there was a significant difference in self-esteem scores between the group-participation adolescents (M = 53.8) and the general population (μ = 50), t(99) = 2.53, p < .05. Additionally, the effect size measure, Cohen's d, revealed a small difference (d = 0.22) in self-esteem scores between the two groups."

To answer these questions, you will need to perform a hypothesis test and compute the effect size using Cohen's d. Here are the step-by-step instructions:

a. Hypothesis Testing:
Step 1: State the hypotheses:
- Null Hypothesis (H0): There is no significant difference between the self-esteem scores of adolescents with group participation and the general population (µ = 50).
- Alternative Hypothesis (Ha): There is a significant difference between the self-esteem scores of adolescents with group participation and the general population (µ ≠ 50).

Step 2: Set the significance level:
Given that the significance level (α) is 0.05, we will divide it equally over both tails (0.025 for each tail).

Step 3: Calculate the test statistic:
The test statistic we will use in this case is the z-score, which is calculated using the formula: z = (M - µ) / (σ / sqrt(n)).
Here, M is the sample mean (53.8), µ is the population mean (50), σ is the population standard deviation (15), and n is the sample size (100).

Step 4: Find the critical value:
Since it is a two-tailed test, we need to find the critical value for α/2 = 0.025 level of significance. This can be found using a Z-table or a statistical calculator.

Step 5: Make a decision:
If the calculated z-score falls within the critical region (outside the critical value), we reject the null hypothesis. Otherwise, if it falls within the non-critical region, we fail to reject the null hypothesis.

b. Effect Size Calculation:
To measure the effect size, we will use Cohen's d, which is calculated using the formula: d = (M - µ) / σ.
Here, M is the sample mean (53.8), µ is the population mean (50), and σ is the population standard deviation (15).

c. Writing a research report sentence:
Based on the results of the hypothesis test and the measure of effect size, you can write a sentence for a research report. For example:
"The hypothesis test revealed that there is a significant difference in the self-esteem scores of adolescents with group participation (M = 53.8) compared to the general population (µ = 50), t(99) = X.XX, p < .05. Cohen's d indicated a small effect size (d = X.XX), suggesting a modest difference in self-esteem scores between the two groups."

Please note that you will need to fill in the appropriate values calculated in your analysis.