Question

Childhood participation in sports, cultural groups, and youth groups appears to be related to improved self-esteem for adolescents. 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 μ=40 and a standard deviation of σ=12. The sample of group-participation adolescents had an average of M=43.84.

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 α =0.01. b. Write a sentence describing the outcome of the hypothesis test as it would appear in a research report.

Answer 1

The provided sample mean is Xˉ=43.84 and the known population standard deviation is σ=12, and the sample size is n=100.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

H0: μ=40

Ha: μ≠​40

This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.01, and the critical value for a two-tailed test is zc​=2.58.

The rejection region for this two-tailed test is R={z:∣z∣>2.58}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that ∣z∣=3.2>zc​=2.58, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0014, and since p=0.0014<0.01, it is concluded that the null hypothesis is rejected.

b) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is different than 40, at the 0.01 significance level.Therefore this sample provide enough evidence to conclude that self-esteem scores for these adolescents are significantly different from 40.