Female undergraduates in randomized groups of 24 took part in a self-esteem stud

Question

Female undergraduates in randomized groups of 24 took part in a self-esteem study. The study measured an index of self-esteem from the point of view of competence, social acceptance, and physical attractiveness. Let x1, x2, and x3 be random variables representing the measure of self-esteem through x1 (competence), x2 (social acceptance), and x3 (attractiveness). Higher index values mean a more positive influence on self-esteem.

Variable Sample Size Mean x Standard Deviation s Population Mean

x1 24 19.06 3.20 ?1

x2 24 19.48 3.98 ?2

x3 24 18.82 3.81 ?3

(a) Find a 95% confidence interval for ?1 ? ?2. (Round your answers to two decimal places.)

lower limit

upper limit

(b) Find a 95% confidence interval for ?1 ? ?3. (Round your answers to two decimal places.)

lower limit

upper limit

(c) Find a 95% confidence interval for ?2 ? ?3. (Round your answers to two decimal places.)

lower limit

upper limit

(d) Comment on the meaning of each of the confidence intervals found in parts (a), (b), and (c). At the 95% confidence level, what can you say about the average differences in influence on self-esteem between competence and social acceptance? between competence and attractiveness? between social acceptance and attractiveness?

Explanation / Answer

a)
z value at 95% = 1.96

CI = (x1 - x2) +/- t *sqrt(s1^2/n1+s2^2/n2)
= ( 19.06 - 19.48)+/- 1.96 *sqrt(3.20^2/24+ 3.98^2/24)
= (-2.4632,1.6232)

The 95% CI is (-2.46,1.62)

b)
z value at 95% = 1.96

CI = (x1 - x3) +/- t *sqrt(s1^2/n1+s3^2/n3)
= ( 19.06 - 18.82)+/- 1.96 *sqrt(3.20^2/24+ 3.81^2/24)
= (-1.7506 , 2.2306)

The 95% CI is (-1.75 , 2.23)

c)
z value at 95% = 1.96

CI = (x2 - x3) +/- t *sqrt(s2^2/n2+s3^2/n3)
= ( 19.48 - 18.82)+/- 1.96 *sqrt(3.98^2/24+ 3.81^2/24)
= (-1.5443,2.8643)

The 95% CI is (-1.54,2.86)

d)
Confidence interval between social acceptance and attractiveness is wider than other two interval

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