Question

A study of parental empathy for sensitivity cues and baby temperament (higher scores mean more empathy) was performed. Let x₁ be a random variable that represents the score of a mother on an empathy test (as regards her baby). Let x₂ be the empathy score of a father. A random sample of 27 mothers gave a sample mean of x₁ = 67.00. Another random sample of 26 fathers gave x₂ = 59.85. Assume that σ₁ = 11.90 and σ₂ = 11.41.

(a) Let μ₁ be the population mean of x₁ and let μ₂ be the population mean of x₂. Find a 99% confidence interval for μ₁ – μ₂. (Use 2 decimal places.)

lower limit
upper limit

(b) Examine the confidence interval and explain what it means in the context of this problem. Does the confidence interval contain all positive, all negative, or both positive and negative numbers? What does this tell you about the relationship between average empathy scores for mothers compared with those for fathers at the 99% confidence level?

Because the interval contains only positive numbers, we can say that the mothers have a higher mean empathy score.

Because the interval contains both positive and negative numbers, we can not say that the mothers have a higher mean empathy score.     

We can not make any conclusions using this confidence interval.

Because the interval contains only negative numbers, we can say that the fathers have a higher mean empathy score.

Answer 1

Part a)

Confidence interval :-
( \bar{X}_{1} - \bar{X}_{2}) \pm Z_{\alpha /2} \sqrt{(\sigma _{1}^{2} / n1) + (\sigma _{2}^{2}/n2)}
Z(α/2) = Z (0.01 /2) = 2.576
( 67 - 59.85 ) \pm Z_{ 0.01/2} \sqrt{( 11.9^{2} / 27 ) + ( 11.41^{2}/26 )}
Lower Limit = ( 67 - 59.85 ) - Z_{ 0.01/2} \sqrt{( 11.9^{2} / 27 ) + ( 11.41^{2}/26 )}
Lower Limit = -1.10
Upper Limit = ( 67 - 59.85 ) + Z_{ 0.01/2} \sqrt{( 11.9^{2} / 27 ) + ( 11.41^{2}/26 )}
Upper Limit = 15.40
99% Confidence interval is ( -1.10 , 15.40 )

Part b)

Because the interval contains both positive and negative numbers, we can not say that the mothers have a higher mean empathy score.