Chapter 8: Confidence Intervals
8.1 Confidence Intervals
Confidence Interval: Mean
A confidence interval is a range of numbers that is likely
contains the true mean.
95% confidence interval for the mean:
P( −1.96 ≤ Z ≤ 1.96) = 0.95
z 0.025 = 1.96
[8.1]
Substitute the z - value based on the standard error of the mean
P( −1.96 ≤
X −µ
≤ 1.96) = 0.95
σ/ n
σ
σ
P( X − 1.96
≤ µ ≤ X + 1.96
) = 0.95
n
n
[8.2]
[8.3]
1.96
σ
n
Lower limit = sample mean – margin of error (
)
Upper limit = sample mean + margin of error
Remember that the sample mean X is normally distributed,
if the sample size n is large (central limit theorem).
A confidence interval for the mean is the sample mean
plus and minus the margin of error ( 誤差範圍，誤差幅度
-margin of error
Lower limit
+margin of error
Sample mean X
Upper limit
Since the sample mean changes every time you redo the
sampling. The confidence interval also changes.
A 95% confidence interval is an interval around the sample
mean and there is a 95% chance that the population mean
is contains in the interval.
).
http://www.answers.com/topic/confidence-interval
95 out of 100 intervals contain the population mean µ
A general expression for the confidence interval for a mean:
σ
σ
X ± zα / 2
(
is the standard error of the means) [8.5]
n
n
where zα / 2 = 1.96 for 95% confidence interval,
and zα / 2 = 2.58 for 99% confidence interval,
α = 0.05 for 95% confidence interval and .005 for 99% C. I.
In Eq. [8.5] we assume we know the population standard
deviation σ. When we do not have σ, we use the sample
standard deviation, s. That is the confidence interval becomes
X ± zα / 2
s
n
[8.7]
Eq. [8.7] works when the sample size is large, n = 30 or larger,
or when the population distribution is normal.
Learning activity 8.1-1 Confidence interval for a mean
• Open kbs.xls!Data.
• Calculate the mean and standard deviation of the Kbs
variable by using AVERAGE() AND STDEV().
• Use Excel to calculate the 95% confidence interval
for the mean.
• Use MegaStat | Descriptive Statistics to calculate the
confidence interval
• See kbs.xls!Solution1 for the solution.
• Calculate 99% confidence interval by using Excel and
with MegaStat
• compare the 95% and 99% confidence intervals
(kbs.xls!Solution1a).
Confidence Interval: Proportion
Calculate the confidence interval for a proportion by
p( 1 - p)
p ± zα /2
n
p( 1 - p)
(
is the standard error for the proportion) [8.8]
n
where p is the sample proportion. We do not know π in [7.5].
Learning Activity 8.1-2 Confidence Interval for a Proportion
• Open kbs.xls!p.
• Use Excel to calculate the 95% confidence interval
for the mean.
• Use MegaStat | Descriptive Statistics to calculate the
confidence interval
• See kbs.xls!Solution2 for the solution.
Note: 95% margin of error is approximately SQRT(1/n)
p( 1-p)
p( 1-p)
= 1.96
≈ 1/ n
n
n
Since sqrt(p(1-p)) is largest when p = 0.5 and
Sqrt(0.5*0.5) = 0.5, 1.96*05 approximately equal to 1.
zα / 2
0.6
sqrt(p(1-p))
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
p
0.6
0.7
0.8
0.9
1
σ
 zα / 2σ 
E = zα / 2 (
) ⇒ n=

n
 E 
2
[8.9]
A company wants to estimate the mean amount each
customer purchases within $2 with 95% confidence. A
small sample indicates that the standard deviation is
about $5.5.
Learning Activity 8.2-1 Sample size for a mean
• Open SampleSize.xls!Start.
• Calculate the sample size by using the Eq. [8.9]
• MegaStat | confidence Intervals | Sample Size – Mean
• See SampleSize.xls!solution1.
Sample Size : Proportion
 zα / 2 
n = π (1 − π )

 E 
2
[8.10]
where π is the population proportion. Since you do no know
the true proportion π, you can estimate π by
taking samples. Or use π = 0.5, since π(1 - π) is largest when
π = 0.5. This will give you a conservative n value.
0.6
sqrt(p(1-p))
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
p
0.6
0.7
0.8
0.9
1
A company wants to estimate with a margin of error of 4%
the proportion of people who will vote for a candidate, using
a 95% confidence level. Since the company does not know
the true proportion, it uses π = 0.5.
Learning Activity 8.2-2 Sample size for a proportion
• Open SampleSize.xls!Start.
• Calculate the sample size by using the Eq. [8.10]
• MegaStat | confidence Intervals | Sample Size – p
• See sampleSize.xls!Solution2.
• See sampleSize.xls!Why p of .5.
Learning Activity 8.B-1 Confidence Interval Simulation
• Open CLT-CI.xls.
You can see that sample means derived from uniform
random number have a near normal distribution.
We calculate the confidence intervals for the mean of 30
uniform distributed random numbers (between 0 and 100).
We repeat such calculation 600 times and get
600 intervals. We can see that about 95%
of 600 intervals contains the true mean of 50.
Note that the variance for uniform random variable is
σ = SQRT((ymax-ymin)2/12) =SQRT((100)^2)/12) = 28.87.
Margin of error = 1.96*28.87/sqrt(30) = 10.33