Beginning Statistics, 2nd Edition
Beginning Statistics, 2nd Edition
2nd Edition
ISBN: 9781932628678
Author: Carolyn Warren; Kimberly Denley; Emily Atchley
Publisher: Hawkes Learning Systems
Question
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Chapter 9.1, Problem 14E
To determine

Construct and interpret a confidence interval for the true difference between the two population means.

Expert Solution & Answer
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Answer to Problem 14E

Solution:

The value of the margin of error is 0.59352 and 95% confidence interval for the true difference between the mean amounts of time spent exercising each week by people who work out in the morning and those who work out in the afternoon or evening at the three health centers is (0.39352,0.79352)

Explanation of Solution

Given Information:

Mean hours of exercise who work out in the morning.

i.e. x¯1=4.1 hours per week.

Mean hours of exercise who work out in the afternoon or evening.

i.e. x¯2=3.7 hours per week.

The Standard deviation for people who exercise in the morning.

i.e. σ1=0.7 hours per week.

Standard deviation for people who exercise in the afternoon or evening.

i.e. σ2=0.5 hours per week.

Number of people who exercise in the morning i.e. n1=49.

Number of people who exercise in the afternoon or evening i.e. n1=54.

Formula used:

The confidence interval for the difference between two population means for independent data sets is given by,

(x¯1x¯2)E<μ1μ2<(x¯1x¯2)+E

Or

((x¯1x¯2)E,(x¯1x¯2)+E)

Where x¯1 and x¯2 are the two sample means,

(x¯1x¯2) is the point estimate for the difference between the population means, μ1μ2, and E is the margin of error and it is given as

E=zα/2σ12n1+σ22n2

Where zα/2 is the critical value for the level of confidence, c=1α, such that the area under the standard normal distribution to the right of zα/2 is equal to α2,

σ1 and σ2 are the two population standard deviations, and

n1 and n2 are the two sample sizes.

Calculation:

The margin of error and it is given as,

E=zα2σ12n1+σ22n2(1)

Since the level of confidence is 95%, then the level of significance is given as,

α=10.95=0.05

The value of z from table at α=0.05 is 1.96.

Substitute the values 4.1 for x¯1, 3.7 for x¯2, 0.7 for σ1, 49 for n1, 0.5 for σ2, 54 for n2 and 1.96 for zα2 in the above equation (1).

E=1.96×(0.7)249+(0.5)254=0.23706

The confidence interval is ((x¯1x¯2)E,(x¯1x¯2)+E), that is,

Confidence interval=((4.13.7)0.23706,(4.13.7)+0.23706)=(0.16294,0.63706)

Interpretation:

The 95% confidence interval for the true difference between the mean amounts of time spent exercising each week by people who work out in the morning and those who work out in the afternoon or evening at the three health centers, ranges from 0.16294 to 0.63706.

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Chapter 9 Solutions

Beginning Statistics, 2nd Edition

Ch. 9.1 - Prob. 11ECh. 9.1 - Prob. 12ECh. 9.1 - Prob. 13ECh. 9.1 - Prob. 14ECh. 9.1 - Prob. 15ECh. 9.1 - Prob. 16ECh. 9.2 - Prob. 1ECh. 9.2 - Prob. 2ECh. 9.2 - Prob. 3ECh. 9.2 - Prob. 4ECh. 9.2 - Prob. 5ECh. 9.2 - Prob. 6ECh. 9.2 - Prob. 7ECh. 9.2 - Prob. 8ECh. 9.2 - Prob. 9ECh. 9.2 - Prob. 10ECh. 9.2 - Prob. 11ECh. 9.2 - Prob. 12ECh. 9.2 - Prob. 13ECh. 9.2 - Prob. 14ECh. 9.2 - Prob. 15ECh. 9.2 - Prob. 16ECh. 9.2 - Prob. 17ECh. 9.2 - Prob. 18ECh. 9.2 - Prob. 19ECh. 9.2 - Prob. 20ECh. 9.2 - Prob. 21ECh. 9.2 - Prob. 22ECh. 9.2 - Prob. 23ECh. 9.2 - Prob. 24ECh. 9.3 - Prob. 1ECh. 9.3 - Prob. 2ECh. 9.3 - Prob. 3ECh. 9.3 - Prob. 4ECh. 9.3 - Prob. 5ECh. 9.3 - Prob. 6ECh. 9.3 - Prob. 7ECh. 9.3 - Prob. 8ECh. 9.3 - Prob. 9ECh. 9.3 - Prob. 10ECh. 9.3 - Prob. 11ECh. 9.3 - Prob. 12ECh. 9.3 - Prob. 13ECh. 9.3 - Prob. 14ECh. 9.3 - Prob. 15ECh. 9.3 - Prob. 16ECh. 9.3 - Prob. 17ECh. 9.3 - Prob. 18ECh. 9.3 - Prob. 19ECh. 9.3 - Prob. 20ECh. 9.3 - Prob. 21ECh. 9.4 - Prob. 1ECh. 9.4 - Prob. 2ECh. 9.4 - Prob. 3ECh. 9.4 - Prob. 4ECh. 9.4 - Prob. 5ECh. 9.4 - Prob. 6ECh. 9.4 - Prob. 7ECh. 9.4 - Prob. 8ECh. 9.4 - Prob. 9ECh. 9.4 - Prob. 10ECh. 9.4 - Prob. 11ECh. 9.4 - Prob. 12ECh. 9.4 - Prob. 13ECh. 9.4 - Prob. 14ECh. 9.4 - Prob. 15ECh. 9.4 - Prob. 16ECh. 9.4 - Prob. 17ECh. 9.4 - Prob. 18ECh. 9.4 - Prob. 19ECh. 9.4 - Prob. 20ECh. 9.4 - Prob. 21ECh. 9.4 - Prob. 22ECh. 9.4 - Prob. 23ECh. 9.4 - Prob. 24ECh. 9.4 - Prob. 25ECh. 9.5 - Prob. 1ECh. 9.5 - Prob. 2ECh. 9.5 - Prob. 3ECh. 9.5 - Prob. 4ECh. 9.5 - Prob. 5ECh. 9.5 - Prob. 6ECh. 9.5 - Prob. 7ECh. 9.5 - Prob. 8ECh. 9.5 - Prob. 9ECh. 9.5 - Prob. 10ECh. 9.5 - Prob. 11ECh. 9.5 - Prob. 12ECh. 9.5 - Prob. 13ECh. 9.5 - Prob. 14ECh. 9.5 - Prob. 15ECh. 9.5 - Prob. 16ECh. 9.5 - Prob. 17ECh. 9.5 - Prob. 18ECh. 9.5 - Prob. 19ECh. 9.5 - Prob. 20ECh. 9.5 - Prob. 21ECh. 9.5 - Prob. 22ECh. 9.5 - Prob. 23ECh. 9.5 - Prob. 24ECh. 9.5 - Prob. 25ECh. 9.5 - Prob. 26ECh. 9.5 - Prob. 27ECh. 9.5 - Prob. 28ECh. 9.CR - Prob. 1CRCh. 9.CR - Prob. 2CRCh. 9.CR - Prob. 3CRCh. 9.CR - Prob. 4CRCh. 9.CR - Prob. 5CRCh. 9.CR - Prob. 6CRCh. 9.CR - Prob. 7CRCh. 9.CR - Prob. 8CRCh. 9.CR - Prob. 9CRCh. 9.CR - Prob. 10CRCh. 9.CR - Prob. 11CRCh. 9.CR - Prob. 12CRCh. 9.CR - Prob. 13CRCh. 9.CR - Prob. 14CRCh. 9.CR - Prob. 15CRCh. 9.P - Prob. 1P
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