Statistical Interval for a Single Sample
Outlines:
 Confidence interval on the mean of a normal
distribution, variance known.
 Confidence interval on the mean of a normal
distribution, variance unknown.
 Confidence interval on the variance and standard
deviation of a normal distribution.
 Large-sample confidence interval for a population
proportion.
Confidence interval



Confidence interval: Bounds represent an interval of plausible
values for a parameter.
Suppose that we estimate the mean viscosity of a chemical
product to be   ˆ  x  1000 , we do not know exactly that the
mean likely to be between 900 and 1100? or 990 and 1010?
Because we use a sample from the population to compute the
interval, we have high confident that it does contain the
unknown population parameter.
Confidence interval

Practical example
A machine fills cups with margarine, and is supposed to be adjusted so that the
mean content of the cups is close to 250 grams of margarine. Of course it is
not possible to fill every cup with exactly 250 grams of margarine. Hence
the weight of the filling can be considered to be a random variable X. The
distribution of X is assumed here to be a normal distribution with unknown
expectation μ and known standard deviation σ = 2.5 grams.
To check if the machine is adequately calibrated, a sample of n = 25 cups of
margarine is chosen at random.

The sample shows actual weights x1 , x 2 , x 3 ,..., x 25 , with mean: x

if the population mean actually around 250g. The value of

If
x  280 . 6 ,   ? ,
population mean shouldn’t close to 250g.

1
n
25
x
i 1
i
 250 . 2
x  250 . 4 , 251 . 1
Confidence interval (Case I)
Confidence interval on the mean of a normal distribution,
variance known.
 Suppose that X1, X2, ...,Xn is a random sample from a normal
distribution with unknown μ and known σ2 .
 We known that
X ~ N ( , / n )
Z 

X 
 /n
A Confidence interval estimate for μ is
L   U
 P{ L    U }  1  
Prob. of selecting samples provide the range of µ that contains the true value of µ
Confidence interval (Case I)

In order to find lower and upper confidence limits:
P{ z  / 2 
P{ X  z / 2
X 
 /

n
n
 z / 2 }  1  
   X  z / 2

n
}  1
Confidence interval (Case I)



Interpreting a CI
We cannot say: "with probability (1 − α) the parameter μ lies in the
confidence interval."
We can say that: if an infinite number of random samples are collected and
a 100(1-)% CI for µ is computed from each sample, 100(1-)% of these
intervals will contain the true value of µ
Confidence interval (Case I)
Ex. Ten measurements of impact energy on specimens of steel are: 64.1, 64.7,
64.5, 64.6, 64.5, 64.3, 64.6, 64.8, 64.2, and 64.3. Assume that impact
energy is normally distributed with σ= 1J. We want to find a 95% CI for μ
That is, based on the sample data, a range of highly plausible values for mean
impact energy for steel is 63.84J-65.08J.
Confidence interval (Case I)

Choice of Sample Size
Confidence interval (Case I)
Ex. Consider the previous example, we want to determine how many specimens
must be tested to ensure that the 95% CI on μ of steel has a length at most
1.0J.
CI length <= 1.0J, E= 0.5J
2
2
z  
 (1 . 96 )1 
n    /2   
  15 . 37
 0 .5 
 E 
n = 16
Confidence interval (Case I)

One-Sided Confidence Bounds
Ex. From previous Ex, find a lower one sided 95% CI for mean impact energy.
x  z 0 .5

 
n
64 . 46  1 . 64
1
10
63 . 94  
 
Confidence interval (Case I)



Large Sample Confidence Interval for μ
X i has any distribution, n>=40, variance unknown
We can approximate CI for μ by replacing σ by S.
Confidence interval (Case I)

Ex
Confidence interval (Case I)
Confidence interval (Case II)
Confidence interval on the mean of a normal distribution,
variance unknown.


Suppose that X1, X2, ...,Xn is a random sample from a normal distribution
with unknown μ and unknown σ2 .
n<40
Confidence interval (Case II)
Confidence interval (Case II)
Confidence interval (Case II)

Ex
Confidence interval (Case II)
Confidence interval (Case III)
Confidence interval on the variance and standard deviation of a
normal distribution.
Confidence interval (Case III)

Two-Sided CI

One-Sided CI
Confidence interval (Case III)

Ex
Confidence interval (Case IV)
Large-sample confidence interval for a population proportion.
Normal approximation for a Binomial Proportion
Confidence interval (Case IV)
Confidence interval (Case IV)

Ex
Confidence interval (Case IV)


Choice of sample size
we are at least 100(1-α)% confident that the error in estimating p bypˆ is
less than E if the sample size is
Confidence interval (Case IV)

Ex.
Confidence interval (Case IV)

One-Sided CI
Homework
1. A confidence interval estimate is desired for the gain in a circuit on a semiconductor device. Assume that gain
is normally distributed with sd.=20.
a)
Find a 95% CI for µ when n =10 and x  1000
b)
Find a 95% CI for µ when n =25and x  1000
c)
Find a 99% CI for µ when n =10 and x  1000
d)
Find a 99% CI for µ when n =25 and x  1000
e)
How does the length of CIs computed above change with the changes in sample size and confidence level?
2. The sugar content of the syrup in canned peaches is normally distributed. A random sampling of n=10 cans
yields a sample standard deviation of s=4.8 mg. Calculate a 95% two-sided confidence interval for 
3. The 2004 presidential election exit polls from the critical state of Ohio provided the following results. There
were 2020 respondents in the exit polls and 768 were college graduates. Of the college graduates, 412
votes for George Bush.
a)
b)
Calculate a 95% confidence interval for the proportion of college graduates in Ohio that voted for George
Bush?
Calculate a 95% lower confidence bound for the proportion of college graduates in Ohio that voted for
George Bush?