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Aim What are confidence intervals for means that have unknown

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					Aim: What are confidence intervals
  for means that have unknown
 standard deviations and sample
       sizes less than 30?
            Quiz Friday
               T distribution
• Used when sample size is less than 30 and the
  variable is normally or approximately normally
  distributed
• Important Characteristics:
  – It is a bell-shape
  – It is symmetric about the mean
  – The mean, median and mode are equal to 0 and
    are located at the center of the distribution
  – The curve never touches the x axis
     How T distribution differs from
      standard normal distribution
• The variance is greater than 1
• The t distribution is actually a family of curves
  based on the concept of degrees of freedom,
  which is related to sample size
• As the sample size increases, the t distribution
  approaches the standard normal distribution
           Degrees of freedom
• Degrees of freedom: the number of values
  that are free to vary after a sample statistics
  has been computed
  – Tells the researcher which specific curve to sue
    when a distribution consists of a family of curves



                    d.f. = n -1
The t family of Curves
    Formula for a specific confidence
         interval for the mean
• Use this when standard deviation is unknown and
  n < 30

                s                  s 
        X  t        X  t 2 
              2 n 
                                        
                                     n


• The values of tα/2 are found in table F. The top
  row of table F, labeled confidence intervals is
  used to get these values
Using the F Table
                  Example
• Ten randomly selected automobiles were
  stopped and the tread depth of eh right front
  tire was measured. The mean was 0.32 inch,
  and the standard deviation was 0.08 inch. Find
  the 95% confidence interval of the mean
  depth. Assume the variable is approximately
  normally distributed.
                       Solution
• Since σ is unknown and s must replace it, the t
  distribution must be used for 95% confidence
  interval. Hence the d.f. = 9, tα/2 =2.262 (from
  table F).
               s                    s 
      X  t 
             2
                        X  t 2      
                 n                    n
                      .08                         .08 
     .32   2.262           .32   2.262       
                      10                          10 
     .32  0.057    .32  .057
     .26    .38
                        Class Work
1. Find the values for each
   1.   tα/2 and n = 18 for the 99% confidence interval for the mean
   2.   tα/2 and n =20 for the 95% confidence interval for the mean
2. A recent study of 25 students showed that they spent an
   average of $18.53 for gasoline per week. The standard
   deviation of the sample was $3.00. Find the 95%
   confidence interval of the true mean.
3. The number of unhealthy days based on the Air Quality
   Index for a random sample of metropolitan areas is
   shown. Construct a 98% confidence interval based on the
   data.
      61      12     6      40     27    38
      93      5      13     40

				
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posted:1/9/2013
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