# Interval Estimation

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```					Interval Estimation
Interval Estimation
 Point Estimate: A single number used to
estimate a population value.
 Interval Estimate: A range within which a
population value probably lies.
 Interval Estimator (Confidence Interval):
A formula that tells us how to use sample
data to calculate an interval that
estimates a population parameter.
We will calculate interval estimates for the
population mean and the population
proportion.
Confidence Interval for the
Population Mean,  .
Confidence Interval for the Population
Mean
   The Confidence Interval (C.I.) for the
population mean is given by:

  
C.I. = x  z 2 

   

 n
   Where z  2 is the z value with an area of
 2 to its right. The parameter  is the
standard deviation of the sampled
population.
Confidence Interval for the Population
Mean
   When  is not known and n is large, the
confidence interval is calculated by:
 s 
x  z 2 
   

 n
Where s is the sample standard deviation.
The confidence coefficient is the probability that a
randomly selected confidence interval will contain
the true population parameter.
The confidence level is the confidence coefficient
expressed as a percentage.
Interpreting a Confidence Interval for
the Population Mean
   When we form a 100(1   )% confidence interval
for the population mean, we usually
express our confidence in the interval by
saying “we can be 100(1   )% confident that 
lies between the lower and upper bounds
of the confidence interval.” This means that
in repeated sampling 100(1   )% of the
resulting intervals will contain  .
Confidence Interval for the Population
Mean
   Example 1: A random sample of size n =
100, from a population with variance  2 =
14400 has mean x = 800. Construct a
95% confidence interval for the population
mean, .
Confidence Interval for the Population
Mean
   Example 2: A biochemist wishes to
estimate the mean vitamin content of a
fruit. From a random sample of size 36, he
calculates the sample mean to be 12.5mg
and the standard deviation to be 3.0mg.
Construct a 99% confidence interval for
the true vitamin content of the fruit.
Confidence Interval for the Population
Mean
   Example 3: Unoccupied seats on flights cause
airlines to lose revenue. Suppose a large airline
wants to estimate its average number of
unoccupied seats per flight over the past year. To
do this, the records of 225 flights are randomly
selected and the number of unoccupied seats is
noted for each of the sampled flights. The sample
mean and standard deviation are 11.6 and 4.1
seats respectively. Estimate the mean number of
unoccupied seats per flight during the past year
using a 90% confidence interval.
Confidence Interval for the
Population Proportion
Confidence Interval for the Population
Proportion
   The C.I. for the population proportion is
given by:           pq              p(1  p) 
 C.I. = p  z 2 
ˆ              = p  z 2 
ˆ                   
 n                  n 
                          
   When p is not known and the sample is
large, the formula is given by:
 p(1  p) 
ˆ     ˆ
p  z 2 
ˆ


    n    
Confidence Interval for the Population
Proportion
    Example 4: A firm is considering
producing and promoting a new product.
It is estimated that the new product will
be profitable if at least 5 percent of the
population buy it within the first year of
its introduction. A survey of 200 people
has shown that 6 people would buy the
product.
i.    Give a point estimate for the proportion of
buyers in the population.
ii.   Construct a 95% C.I. for the true population
proportion.
Confidence Interval for the Population
Proportion
   Example 5: A food products company conducted
a market study by randomly sampling and
interviewing 1000 consumers to determine which
brand of breakfast cereal they prefer. Suppose
313 consumers were found to prefer the
company’s brand. How would you estimate the
true fraction of all consumers who prefer the
company’s cereal brand? Construct a 95%
confidence interval for the proportion of all
consumers who prefer the brand.
Determining the Sample Size
   The appropriate sample size for making an
inference about a population parameter depends on
the desired reliability or the maximum allowable
error.
   The maximum allowable error when forming a
confidence interval for the population mean is given
by:
  
E  z 2 
   

  n
Transposing this formula and solving for n yields:
 z 
2

n 
 E 
Determining the Sample Size
   The maximum allowable error when forming a
confidence interval for the population proportion
is given by:
p (1  p )
ˆ      ˆ
E  z 2
n

Transposing this formula and solving for n yields:
2
z
n    p(1  p)
ˆ     ˆ
E
Determining the Sample Size
 Example 6: Recall example 2, what
sample size is required if the 95% C.I. is
to be done so that the maximum error of
the estimate is 0.5mg?
 Example 7: Recall example 5, suppose the
company wished to estimate its market
share to within 0.015 with a 95% C.I.,
what sample size is required?

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 views: 229 posted: 4/16/2010 language: English pages: 16