CONFIDENCE INTERVAL FOR TWO INDEPENDENT SAMPLE MEANS (σ and σ

Section AA

Confidence Intervals for Difference of two Means

Two Designs for determining confidence intervals for the difference of two means:

Independent Samples: Two samples are independent if the observations in one sample do not influence the

observations in the other.

Paired Samples: Two samples are paired if each observation in one sample can be paired with an observation

in the other. Typically the samples consist of pairs of measurements on the same individual, or pairs of

individuals who are related, such as twins, husbands and wives or brothers and sisters.

Independent Samples

Assumptions:

1) We have simple random samples from two populations.

2) The samples are independent of one another.

3) Each sample size is large (n > 30), or its population is approximately normal.

Point estimate for the difference of two means, ��1 �C ��2 is given by: x1 ? x 2

Standard error of x1 ? x 2 =

Standard error of x1 ? x 2 =

? 12

n1

?

? 22

n2

if the population standard deviations are known.

s12 s 22

?

if the population standard deviations are unknown.

n1 n2

Degrees of freedom = smaller of n1 �C 1 and n2 �C 1

(Note: Calculating the degrees of freedom using technology is different than by hand.)

CONFIDENCE INTERVAL FOR TWO INDEPENDENT SAMPLE MEANS (��1 and ��2 known)

( x1 ? x2 ) ? z?

? 12

2

n1

?

? 22

n2

CONFIDENCE INTERVAL FOR TWO INDEPENDENT SAMPLE MEANS (��1 and ��2 unknown)

( x1 ? x2 ) ? t?

2

s12 s 22

?

n1 n2

df = smaller of n1 �C 1 and n2 �C 1

31

Examples

1) In order to determine whether or not there is a significant difference between the hourly wages of two

companies, the following data have been accumulated:

Company A : n = 80, x = $6.75 and �� = $1.00

Company B : n= 60, x = $6.25 and �� = $0.95

Find a 95% confidence interval for the population mean difference between the hourly wages of the two

companies. Interpret this result. Is there a difference between the two companies hourly wage?

95% CI �� z = 1.96

??

?. ?? ? ?. ?? �� ?. ??��?? +

?.???

�� (0.17, 0.83)

??

I am 95% confident the population mean difference is between $0.17 and $0.83.

There is a difference between the two companies.

2) A teacher desires to estimate the difference in reading levels between children from two-parent homes

and children from one-parent homes in his community. Using 2 random samples of fourth-grade youngsters,

he finds that 19 children from two-parent homes had a mean reading level of 5.1 with a standard deviation of

1.4 and that 13 children from one-parent homes had a mean reading level of 3.8 with a standard deviation of

2.1. Find a 98% confidence interval for the population mean difference in the reading levels. Interpret this

result. Is there a difference between two-parent and one-parent homes?

2-parent 1-parent

n = 19

n = 13

? = 5.1

? = 3.8

?

?

s = 1.4

s = 2.1

98% CI �� t = 2.681

df = 12

?.??

?. ? ? ?. ? �� ?. ???�� ?? +

?.??

??

�� (?0.48, 3.08)

I am 98% confident the population mean difference is between ?0.48 and 3.08.

There is no difference between 2- and 1-parent homes.

3) The Gypsy Taxi Cab Company of Brooklyn, New York, is still checking tires. They now want to know if cab

drivers under 25 are harder on tires than are older drivers. Of their 66 cabs, 27 are driven exclusively by the

younger drivers and the remaining 39 by the older drivers. The younger drivers average 17,482 miles for a set

of tires with a standard deviation of 1320, while the older drivers average 17,728 miles with a standard

deviation of 981 miles. Find a 90% confidence interval for the population mean difference between the miles

driven for older and younger drivers. Interpret this result. Is there a difference between older and younger

drivers?

Under 25 Over 25

n = 27

n = 39

? = 17482 ?

? = 17728

?

s = 1320

s = 981

90% CI �� t = 1.706

df = 26

?????

????? ? ????? �� ?. ???��

??

+

????

??

�� (?755.55, 263.55)

I am 90% confident the population mean difference is between ?755.55

and 263.55 miles.

There is no difference between younger and older drivers.

32

4) Gotcha, the consumer magazine, is testing the lives of 2 kinds of flashlight batteries. Britelite claims to give

more life in normal use but is more costly than ordinary batteries. In a random sample of 37 Britelite

batteries, mean life was 17.5 months of normal use with a standard deviation of 1.1 months. In a random

sample of 28, ordinary batteries, the mean life was 14.7 months with a standard deviation of 1.3 months. Find

a 95% confidence interval for the population mean time difference in the lives of the batteries. Interpret this

result. Is there a difference between the two batteries?

Britelite

Ordinary

n = 37

n = 28

? = 17.5

? = 14.7

?

?

s = 1.1

s = 1.3

95% CI �� t = 2.052

df = 27

?.??

??. ? ? ??. ? �� ?. ???�� ?? +

?.??

??

�� (2.17, 3.43)

I am 95% confident the population mean difference is between 2.17 and

3.43 months.

There is a difference between Britelite and ordinary batteries.

5) The concentration of benzene was measured in units of milligrams per liter for a simple random sample of

five specimens of untreated wastewater produced at a gas field. The sample mean was 7.8 with a standard

deviation of 1.4. Seven specimens of treated wastewater had an average benzene concentration of 3.2 with a

standard deviation of 1.7. It is reasonable to assume that both samples come from populations that are

approximately normal. Construct a 99% confidence interval for the population mean difference in the benzene

concentration. Interpret this result. Is there a difference between the untreated and treated wastewater?

Does the treatment work?

Untreated Treated

n=5

n=7

? = 7.8

? = 3.2

?

?

s = 1.4

s = 1.7

99% CI �� t = 4.604

df = 4

?.??

?. ? ? ?. ? �� ?. ???��

?

+

?.??

?

�� (0.47, 8.73)

I am 99% confident the population mean difference is between 0.47 and 8.73

milligrams per liter.

There is a difference between untreated and treated wastewater.

Yes, the treatment works.

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