Confidence_intervals

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					Confidence intervals
    Summer program
      Brian Healy
                  Last class
 Central limit theorem
 Hypothesis testing
    – Null and Alternative hypotheses
    – Test statistic
    – p-value
    – Conclusion
    What are we doing today?
 Confidence interval
 Comparison between confidence interval
  and hypothesis testing
 Practice problems
 How to do this in R
 R functions
         Steps for hypothesis testing
1)       State type of test and alpha level
2)       State null and alternative hypotheses
3)       Determine and calculate appropriate test
         statistic
4)       Calculate p-value
5)       Decide whether to reject or not reject the null
         hypothesis
     •     NEVER accept null
6)       Write conclusion
                         Example
   A former student of mine collected a large amount of
    demographic data from school children in Afghanistan.
    Since this population was possibly malnourished, she
    was concerned that the children would have a
    hemoglobin level below the healthy average. The
    healthy average is 13 g/dL.
   She asked me to run a hypothesis test comparing the
    hemoglobin levels in her sample population to the
    healthy average value. She had collected a sample of
    size 127 children.
   Sample hemoglobin levels:
    – Mean = 11.7 g/dL
    – Standard deviation = 1.2 g/dL
           Steps for hypothesis testing
1)       We are doing a one-sample test with alpha=0.05
2)       Hypotheses
     •     H0: m=13 g/dL
     •     HA: m != 13 g/dL
3)                  11 .7  13
     t dof 126                12 .4
                    1.2 127

4)       p-value < 0.0001
5)       Reject null hypothesis
6)       Conclusion: There is a significant difference between
         the average hemoglobin levels in the children in
         Afghanistan and the normal average hemoglobin level
              Something more
 Up to this point we have drawn a sample and estimated
  the population value with the sample mean. This was
  called a point estimate.
 Beyond the simple point estimate, we have used the
  standard deviation of the sample to allow us to test
  hypotheses about the population mean. The only
  problem was that we could only answer Yes or No about
  the hypothesis test.
 Now, we may want to know even more than the point
  estimate and the specific hypothesis test results so that
  we know an interval of plausible values for the
  population mean based on our sample
 Confidence interval
             Confidence interval
   Definition: a set of values that we believe are plausible
    estimates of population mean based on the sample we
    have drawn
   As we discussed yesterday, when we take multiple
    samples, the sample mean will not be the same every
    time (in fact it will almost certainly be different). The
    confidence interval is an interval around our sample
    mean that allows us to have a certain amount of
    confidence that the true mean is covered by the interval.
   We can draw conclusions about the true population
    mean based on our confidence interval
                       Example
   Although the hypothesis test for our children told us that
    the average hemoglobin level is lower than the average
    level in the United States, we have learned very little
    about the actual health of the children. Beyond the point
    estimate of the population mean, my former student was
    interested in knowing what was the plausible range of
    values for the population mean hemoglobin level
    because she knows that her sample mean is a function
    of her sample. Basically, she would like to know, “How
    accurate was this 11.7 value?”. This would allow her to
    see how the children compare to other countries and
    decide on possibe interventions.
       Construction of a confidence
                 interval
   To construct a confidence interval we need to go back to
    probability…
   We are going to start with a standard normal. Remember
    that 1.96 leaves 0.025 in the upper tail in a standard
    normal RV. Note that we are using the population
    variance .
    P 1.96  Z  1.96   0.95
               X m
    P (1.96        1.96 )  0.95
                n
                                       
    P ( X  1.96        m  X  1.96       )  0.95
                   n                    n
   The probability statement now says something about m,
    but remember m is not a random variable
   The resulting interval we get is, which means that we are
    95% confident that this interval will cover m
                            
     X  1.96    , X  1.96   
               n             n
   We must be careful about the interpretation:
    – This does NOT mean “m falls within this interval 95% of the time”
      or “95% of the population values lie between these limits” or
      “there is a 95% chance that m is in the interval” because m is a
      specific value
    – This does mean “if we selected 100 random samples from the
      population and calculated 100 confidence intervals for m,
      approximately 95 of the intervals would cover m and 5 would not
   A more general confidence interval is
                      
     X  z 2   , X  z 2  
              n            n
              Illustration
 Let’s look at this through a simulation
 www.
 Note that we do not always cover the
  population mean exactly 95 times out of
  100, but on average we will.
 What can you say about the 95% and
  99% confidence intervals?
        Changing the width of the
           confidence interval
   The width of the confidence interval is based on
    3 factors
    – confidence level (z)- how confident do we want to be
      that the interval covers m; the higher the confidence,
      the wider the interval
    – variance )- how different might the samples be; the
      more variability, the wider the interval
    – sample size (n)- how many samples did we use to
      estimate the population mean; the larger the sample,
      the better the point estimate, the narrower the
      interval
                             Practice
   We would like to provide a 95% confidence
    interval for the hemoglobin level for the children
    in the school. Assume we know the population
    variance is equal to the sample variance
                1.2              1.2 
    11.7  1.96     ,11.7  1.96       (11.49,11.91)
                127              127 

   For a 99% interval,
                1.2              1.2 
    11.7  2.58     ,11.7  2.58       (11.43,11.97)
                127              127 
                    Conclusions
   We are 95% confident that the true mean level of
    hemoglobin in school children is between 11.49 and
    11.91. Beyond that, we are 99% confident that the true
    mean level is between 11.43 and 11.97.
   We cannot say that there is a 95% chance that the true
    mean level of hemoglobin is between and because either
    the true mean is in the interval or not.
   Remember that our confidence interval is subject to the
    same sampling variability as the hypothesis test in that
    sometimes just by chance our confidence interval will
    not cover the true population mean.
    One-sided confidence interval
 These are very common in everyday life
  (catching a bus or train), but far less common in
  statistical applications
 These are either a lower (upper) bound because
  instead of being 95% confident that the mean is
  in an interval, we now say that we are 95%
  confident that the mean is above (below) a
  given value. To have 0.05 in the lower (upper)
  tail, the cut-off from the standard normal
  distribution is
   -1.645 (1.645). How could we have found this
  value in R?
           One-sided continued
   The 95% one-sided confidence interval (lower
    bound) is  X  1.645  ,  
                               
                       n   

   The 95% one-sided confidence interval (upper
    bound) is   , X  1.645  
                                
                           n

   The interpretation of these are “we are 95%
    confident that m is at most (at least) the given
    upper bound (lower bound).
      Confidence interval with the t-
               distribution
   As we discussed yesterday, often we do not know the
    population standard deviation. In these cases, we need
    to use the sample standard deviation and the t-
    distribution.
   The entire procedure for finding the confidence interval is
    the same as for the normal confidence interval, but the
    cut-offs are from the t-distribution. Remember the
    degrees of freedom for the t-distribution are the total
    sample size minus 1 (n-1)
                   s                 s 
      X  tn1, 2    , X  tn1, 2   
                    n                 n
        Confidence intervals in R
   Anytime you perform a hypothesis test in R a
    confidence interval is given as well
    – t.test(vector, [alternative=], conf.level=.95)
    – The confidence interval will be one-sided or two-sided
      based on the alternative
    – Remember that the default hypothesis test is that
      m=0, so the p-value you get may not be relevant
   Practice: write a function that allows the user to
    input a sample vector, the population standard
    deviation, and the confidence level and outputs
    a normal two-sided confidence interval
                             Example
   Let’s build a confidence interval for our hemoglobin using t.test
     – hemolevel<-read.table(“g:\\ \\hemolevel.dat”, header=T)
     – t.test(hemolevel)
             One Sample t-test
     data: hemolevel
     t = 111.4219, df = 125, p-value < 2.2e-16
     alternative hypothesis: true mean is not equal to 0
     95 percent confidence interval:
      11.49530 11.91105
     sample estimates:
     mean of x
      11.70317
   Now, let’s build a one-sided 99% lower confidence interval
     – t.test(hemolevel, alternative=“greater”, conf.level=.99)
     t = 111.4219, df = 125, p-value < 2.2e-16
     alternative hypothesis: true mean is greater than 0
     99 percent confidence interval:
      11.45565      Inf
    Comparison of hypothesis testing
       and confidence interval
   Let’s try a couple of hypothesis tests for
    our hemoglobin level
    – t.test(hemolevel, mu=13): our test from
      before
    – t.test(hemolevel, mu=12)
    – t.test(hemolevel, mu=11.9)
    – t.test(hemolevel, mu=11.91105)
    – What happens to the p-value in each of these
      cases?
   Conclusions?
 As you can see, if you would reject a specific
  null hypothesis H0: mm0, this value is not
  included in the confidence interval. Therefore,
  you can use a confidence interval to test a
  hypothesis just as well as you use a hypothesis
  test.
 The reason for this relationship is because a
  confidence interval is the inversion of the
  hypothesis test, meaning that the confidence
  interval could have been constructed by finding
  all of the values of m for which the hypothesis
  test would fail to reject.
       Possible function for normal
           confidence interval
normalci<-function(data, stand, level){
n<-nrow(data)
xbar<-mean(data)
zalpha<- -qnorm((1-level)/2)
ll<-xbar-zalpha*(stand/sqrt(n))
ul<- xbar+zalpha*(stand/sqrt(n))
list(lowerlim=ll,upperlim=ul)
}

> normalci(hemolevel,2,0.95)
$lowerlim
hemoglobin
11.35396
$upperlim
hemoglobin
  12.05239

				
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