Question

A random sample of 40 adults with no children under the age of 18 years results in a mean daily leisure time of 5.83 ​hours, with a standard deviation of 2.49 hours. A random sample of 40 adults with children under the age of 18 results in a mean daily leisure time of 4.47 ​hours, with a standard deviation of 1.89 hours. Construct and interpret a 95​% confidence interval for the mean difference in leisure time between adults with no children and adults with children (mu 1 - mu 2)

Let mu 1μ1 represent the mean leisure hours of adults with no children under the age of 18 and mu 2μ2

represent the mean leisure hours of adults with children under the age of 18.

The 95​% confidence interval for left parenthesis mu 1 minus mu 2 right parenthesisμ1−μ2 is the range from   

hours to hours.

​(Round to two decimal places as​ needed.)

What is the interpretation of this confidence​ interval?

A.There is 95​% confidence that the difference of the means is in the interval. Conclude that there is insufficient evidence of ainsufficient evidence of asignificant difference in the number of leisure hours.

B.There is a 95​%probability that the difference of the means is in the interval. Conclude that there is a significant difference in the number of leisure hours.

C.There is 95​% confidence that the difference of the means is in the interval. Conclude that there is a significant difference in the number of leisure hours.

D. There is an 95​%probability that the difference of the means is in the interval. Conclude that there is insufficient evidence of a significant difference in the number of leisure hours.

Answer 1

We have to conduct t test to find the result. The formula for the test below.

here for sample 1

X1bar=5.83

Sd1=2.49

N1=40

Sample 2

X2bar=4.47

Sd2=1.89

N2=40

For 95% conf interval t stat will be at df=N1+N2-2 = 40+40-2=78 from t table

t=-/+1.990847

-/+1.990847=((5.83-4.47)-(mu1-mu2))/sqrt(2.49^2/40+1.89^2/40)

solving we get

mu1-mu2=2.34402

mu1-mu2=0.3759797

The 95​% confidence interval for left parenthesis mu 1 - mu 2 (μ1−μ2) is the range from 0.3759797 to 2.34402

Since the interval doen't consist value 0 that mean if the interval has value 0 them mu1-mu2=0 hence mu1=mu2 we could have concluded but the interval doesn't contain 0 hence we can conclude as.,

Ans=> There is a 95​%probability that the difference of the means is not in the interval. Conclude that there is a significant difference in the number of leisure hours.

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