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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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