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 = 69.89. Another random sample of 40 fathers gave x2 = 61.55. Assume that ?1 = 12.04 and ?2 = 11.20.
(a) Let ?1 be the population mean of x1 and let ?2 be the population mean of x2. Find a 99% confidence interval for ?1 – ?2. (Use 2 decimal places.)
(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?
a. Because the interval contains only positive numbers, we can say that the mothers have a higher mean empathy score.
b. Because the interval contains both positive and negative numbers, we can not say that the mothers have a higher mean empathy score.
c. We can not make any conclusions using this confidence interval.
d. Because the interval contains only negative numbers, we can say that the fathers have a higher mean empathy score.
x1(bar) = 69.89
x2(bar) = 61.55
s1 = 12.04
s2 = 11.20
n1 = 27
n2 = 40
SE = sqrt[ (s12/n1) + (s22/n2) ]
(s12/n1) = 5.3689
(s22/n2) = 3.1360
SE = 2.9163
DF = 27 - 1 = 26
x1bar - x2bar = 8.34
SE = 2.9163
CI = 99%
DF = 26
t-value = 2.7787
ME = t*SE = 2.7787*2.9163 = 8.1036
Confidence Interval is (x1bar - x2bar) +/- ME
Lower bound = 8.34 - 8.1036 = 0.23637
Upper bound = 8.34 + 8.1036 = 16.44363
Confidence Interval (0.2364 , 16.4436 )
Because the interval contains only positive numbers, we can say that the mothers have a higher mean empathy score.
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