Question

One of the most widely administered psychological assessments is the Rosenberg Self-Esteem Scale (1965), a 10-item scale that assesses global self-worth by measuring both positive and negative feelings about the self. Scores on this standardized self-esteem questionnaire range from 10-100, with higher scores indicating higher levels of self-esteem. For the general population of American teenagers, the average score is 50 points (SD = 15 points).

Childhood participation in organizations such as sports, cultural groups, religious groups, and school clubs may be related to improved self-esteem for teens (McGee et al, 2016). To test these ideas, researchers conducted three studies. In each one, a sample of teens that had participated in one of these types of organizations was collected, and each participant took the Rosenberg Self-Esteem Scale. Using the information provided below for each study, state the sample mean (M), the population mean (μ), the sample size (n), the population standard deviation (σ) and the standard error (σ_M). Then, answer the four questions about the Z scores for the sample mean and the percentiles and probabilities associated with it. (Note- these problems are very similar to those demonstrated in the textbook chapter).

Study 2: A sample of 25 teens with a history of childhood participation in a cultural organization are given the Rosenberg Self-Esteem Scale. Their average score is 54 points.

M = μ = n = σ = _____________ σM = _____________

5. Find the z-score for this sample mean.

6. At what percentile is this sample mean?

7. What is the probability of selecting a sample whose mean self-esteem score is greater than this sample?

8. What is the probability of selecting a sample whose mean self-esteem score is less than this sample?

Study 3: A sample of 20 teens with a history of childhood participation in a religious club are given the Rosenberg Self-Esteem Scale. Their average score is 48 points.

M = μ = n = σ = _____________ σM = _____________

9. Find the z-score for this sample mean.

10. At what percentile is this sample mean?

11. What is the probability of selecting a sample whose mean self-esteem score is greater than this sample?

12. What is the probability of selecting a sample whose mean self-esteem score is less than this sample?

Answer 1

A sample of 25 teens with a history of childhood participation in a cultural organization are given the Rosenberg Self-Esteem Scale. Their average score is 54 points.

M = 54

μ = 50

n = 25

σ =15

σM = 15/sqrt(25)=15/5=3

5. Find the z-score for this sample mean.

z=xbar-mu/sigma/sqrt(n)

=54-50/3

= 1.333333

6. At what percentile is this sample mean?

P(Z<1.333)

From standard normal table z<1.333=0.9082

=0.9082

=0.9082*100

=90.82%

7. What is the probability of selecting a sample whose mean self-esteem score is greater than this sample?

P(X>54)

P(Z>54-50/15)

P(Z> 0.2666667)

=1-P(Z<0.2667)

=1-0.6064

=0.3936

0.3936

8. What is the probability of selecting a sample whose mean self-esteem score is less than this sample?

P(X<54)

P(Z<54-50/15)

=P(Z<0.2667)

=0.6064

0.6064