CONFIDENCE INTERVAL FOR TWO INDEPENDENT SAMPLE MEANS (σ and σ
嚜燙ection 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, 米1 每 米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 每 1 and n2 每 1
(Note: Calculating the degrees of freedom using technology is different than by hand.)
CONFIDENCE INTERVAL FOR TWO INDEPENDENT SAMPLE MEANS (考1 and 考2 known)
( x1 ? x2 ) ? z?
? 12
2
n1
?
? 22
n2
CONFIDENCE INTERVAL FOR TWO INDEPENDENT SAMPLE MEANS (考1 and 考2 unknown)
( x1 ? x2 ) ? t?
2
s12 s 22
?
n1 n2
df = smaller of n1 每 1 and n2 每 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 考 = $1.00
Company B : n= 60, x = $6.25 and 考 = $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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