Question

Female undergraduates in randomized groups of 10 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 10 19.62 3.41 μ1 x2 10 18.99 3.56 μ2 x3 10 18.26 3.46 μ3 (a) Find a 95% confidence interval for μ1 − μ2. (Round your answers to two decimal places.) lower limit upper limit (b) Find a 95% confidence interval for μ1 − μ3. (Round your answers to two decimal places.) lower limit upper limit (c) Find a 95% confidence interval for μ2 − μ3. (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? This answer has not been graded yet.

Answer 1

x1: SAMPLE SIZE = 10, MEAN = 19.62, STANDARD DEVIATION =3.41 .............μ1

x2: SAMPLESIZE = 10, MEAN= 18.99, STANDARD DEVIATION =3.56....................μ2

x3 : SAMPLE SIZE =10,MEAN = 18.26, STANDARD DEVIATION = 3.46..................μ3.

a) 95 % CONFIDENCE INTERVAL FOR μ1 AND μ2

μ₁ - μ₂ = (M₁ - M₂) ± ts_{(M1 - M2)}

M₁ & M₂ = sample means
t = t statistic determined by confidence level = 2.101
s_{(M1 - M2)} = standard error = √((s²_p/n₁) + (s²_p/n₂))

Pooled Variance
s²_p = ((df₁)(s²₁) + (df₂)(s²₂)) / (df₁ + df₂) .... S1=3.41, S2=3.56

= 218.72 / 18

s²_p = 12.15

Standard Error
s_{(M1 - M2)} = √((s²_p/n₁) + (s²_p/n₂))

= √((12.15/10) + (12.15/10)) = 1.56

Confidence Interval
μ₁ - μ₂ = (M₁ - M₂) ± ts_{(M1 - M2)}

= 0.63 ± (2.1 * 1.56)

= 0.63 ± 3.2751

lower limit = -2.645, upper limit =3.9051

b) 95 % CONFIDENCE INTERVAL FOR μ1 AND μ3

μ₁ - μ3 = (M₁ - M3) ± ts_{(M1 - M3)}

M₁ & M₃ = sample means
t = t statistic determined by confidence level
s(M₁ - M₃) = standard error = √((s²_p/n₁) + (s²_p/n₃))

Pooled Variance
s²_p = ((df₁)(s²₁) + (df3)(s₃²)) / (df₁ + df ₃)

= 212.4 / 18 = 11.8

Standard Error
s(M₁ - M₃) = √((s²_p/n₁) + (s²_p/n₃))

= √((11.8/10) + (11.8/10)) = 1.54

Confidence Interval
μ₁ - μ₂ = (M₁ - M3) ± ts(M₁ - M₃) = 1.36 ± (2.1 * 1.54)

= 1.36 ± 3.2275

lower limit = -1.86 upper limit = 4.5875

c) 95 % CONFIDENCE INTERVAL FOR μ2 AND μ3

μ2 - μ3 = (M2 - M3) ± ts_{(M2 - M3)}

M2 & M3 = sample means
t = t statistic determined by confidence level
s_{(M2 - M3)} = standard error = √((s²_p/n2) + (s²_p/n3))

Pooled Variance
s²_p = ((df2)(s₂²) + (df3)(s₃²)) / (df2 + df3) = 221.81 / 18 = 12.32

Standard Error
s_{(M2- M3)} = √((s²_p/n2) + (s²_p/n3))

= √((12.32/10) + (12.32/10)) = 1.57

Confidence Interval
μ₁ - μ₂ = (M2- M3) ± ts_{(M2- M3)}

= 0.73 ± (2.1 * 1.57)

= 0.73 ± 3.2982

lower limit =-2.568 upper limit= 4.0282

a) lower limit = -2.645, upper limit =3.9051.. between competeence and social acceptence

t-statistic	-0.404
DF	18
Significance level	P = 0.6909

b) lower limit = -1.86 upper limit = 4.5875 ..between competeence and attarctiveness

t-statistic	-0.885
DF	18
Significance level	P = 0.3877

c) lower limit =-2.568 upper limit= 4.0282..between social acceptence and attractiveness

t-statistic	-0.465
DF	18
Significance level	P = 0.6475

based on t value and p-value we can say that the average difference is not influenced on the self-esteem of any group.