1. Chapter 8: 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:
Lower limit = sample mean – margin of error ( )
Upper limit = sample mean + margin of error
Remember that the sample mean is normally distributed,
if the sample size n is large (central limit theorem).

2. 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 margin of error
Upper limit
Lower limit
Sample mean
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.

3. 95 out of 100 intervals contain the population mean 
95 out of 100 intervals contain the population mean 

4. A general expression for the confidence interval for a mean:
In Eq. [8.5] we assume we know the population standard
deviation . When we do not have , we use the sample
standard deviation, s. That is the confidence interval becomes
Eq. [8.7] works when the sample size is large, n = 30 or larger,
or when the population distribution is normal.

5. 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).

6. Confidence Interval: Proportion
Calculate the confidence interval for a proportion by
where p is the sample proportion. We do not know  in [7.5].
Learning Activity 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.

7. Note: 95% margin of error is approximately SQRT(1/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.

8. Sample Size Estimation
Sample Size: Mean
We specify the margin of error E (from [8.5])
z/2 = 1.96 for 95% and 2.58 for 99%.
 = population standard deviation.
E = Error tolerance.
We have to estimate .

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 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.

10. Sample Size : Proportion
where  is the population proportion. Since you do no know
the true proportion , you can estimate  by
taking samples. Or use  = 0.5, since (1 - ) is largest when
 = 0.5. This will give you a conservative n value.

11. 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  = 0.5.
Learning Activity 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.

12. 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
 = SQRT((ymax-ymin)2/12) =SQRT((100)^2)/12) =
Margin of error = 1.96*28.87/sqrt(30) = 10.33