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									              Analysis of Means

               Farrokh Alemi, Ph.D.
                Kashif Haqqi M.D.


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                       Table of Content
         •    Review                  •   Use of Z Values
         •    Objectives              •   Confidence Interval
         •    Definitions             •   Hypothesis
         •    Expected Value          •   Two Types of Error
                                      •   One-tailed Tests
         •    Normal Distribution
                                      •   Steps in Testing a
         •    Distribution of Mean        Hypothesis
         •    Central Limit Theorem   •   When to Assume
         •    Standard Normal             Normal Distribution for
              Distribution                Means
                                      •   Use t-distribution
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                           Review
         •    Frequency distribution
         •    Mean, median, and mode
         •    Standard deviation and range


                                             Statistics is the
                                             art of making
                                                sense of
                                             distributions.
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                          Objectives
         •    Describe different distributions, including
              normal, and t-distributions.
         •    Calculate and interpret confidence
              intervals using normal distributions.
         •    Understand types of errors that occurs
              with hypothesis testing.
         •    Hypothesis testing using t-distribution.
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              Example You Should Be Able to
                   Answer at the End
         •     The cost of rehabilitation in the industry is
               $25,000, with a standard deviation of
               3000.
         •     Assume that the average cost in our
               hospital is $30,000.
         •     With 95% confidence, would you say that
               our cost is different than the industry?

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                                    Is it important to ask these
                                         types of questions?
                         Definitions
         •    A random variable is a variable whose
              values are determined by chance.
         •    A probability distribution is the
              probability with which values of a random
              variable can or are observed.
         •    Probability of a value is the frequency of
              occurrence of that value divided by the
              frequency of occurrences of all values.
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          Example of Probability Estimates
         •    We examined the waiting time of 50 people at
              our emergency room and found that 10 people
              waited up to 5 minutes, 20 people waited 5.001
              to 10 minutes, 13 people waited 10.001 to 15
              minutes and 7 people waited 15.001to 20
              minutes.
         •    What is the probability of waiting 5 minutes?
         •    What is the probability of waiting up to 10
              minutes?       Distributions help us make probability
Go to Index                      estimates about observed values.
          Example of Probability Estimates
                   (Continued)
         •    The probability of waiting up to 5 minutes
              is the number of times people waited up to
              5 minutes divided by the total number of
              people: 10/50=.20.
         •    The probability of waiting up to 10
              minutes is the number of people who
              waited up to 10 minutes divided by the
              total number of people: (10+20)/50=0.6.
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                        Expected Value
         •    Expected value of a distribution is the mean of
              the distribution.
         •    It represents our long run expectations about the
              distribution.
         •    The expected value of X is given by summing
              the product of each value of X, referred to as “i”,
              times its probability of occurring, referred to as
              p(X=i).
         •    Expected value = mean =  p(X=i) * i.
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          Example Calculation of Expected
                  Value or Mean
         •    We examined the waiting time of 50
              people at our emergency room and found
              that 10 people waited up to 5 minutes, 20
              people waited 6 to 10 minutes, 13 people
              waited 11 to 15 minutes and 7 people
              waited 16-20 minutes.
         •    What is the mean waiting time at our
              emergency room?
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          Example Calculation of Expected
            Value or Mean (Continued)
                             Observed                          Probability
                             waiting                           times waiting
                             time        Frequency Probability time
                                     2.5        10       0.2              0.5
                                     7.5        20       0.4                3
                                   12.5         13      0.26             3.25
                                   17.5          7      0.14             2.45
         Total                                  50           1            9.2

                             The expected value or mean is 9.2
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              Do this in Excel
                   Normal Distribution
         •    A symmetric distribution, meaning that
              data are evenly distributed about the
              mean.
         •    Mean, median and mode are the same
              value.
         •    It has one mode and looks like a bell
              shaped curve.

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              Normal Distribution Continued
         •     The curve is continuous, there are no gaps
               or holes.
         •     The curve never touches the X-axis as any
               value is possible but with infinitely small
               probabilities.
         •     99.7% of values are within 3 standard
               deviations of mean.

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                   Distribution of Mean
         •    If you take a repeated sample of some
              observations and average them, then you have a
              distribution for the mean.
         •    The distribution of the mean has the same mean
              as the distribution of the observations.
         •    Standard deviation of the mean = Standard error
              = Standard deviation of the observations /
              Square root of the sample size.

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                            Example
         •    What is the mean, standard deviation and
              standard error for the following data: 4, 5,
              6?
         •    Mean = 5
         •    Standard deviation = 1
         •    Standard error = 1 / 1.7 = 0.58

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                          Central Limit Theorem
         •      For any distribution of n observations with
                mean of  and standard deviation .
         •      As n increases, the sample means will
                have a Normal distribution of mean  and
                standard deviation  / square root (n).

                                 The theorem is important because it
                                 helps us ignore questions about the
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              Do this in Excel   mean and standard deviation of it.
              Standard Normal Distribution
         •    A Normal distribution.
         •    Mean of zero.
         •    Standard deviation of 1.
         •    Z = (Observed value – mean) / standard
              deviation of average.
         •    Where standard deviation of mean = standard
              error = standard deviation of observations
              divided by square root of sample size.
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               Example Calculation of Z
         •    What is the Z value for the observed mean
              of 16, if the average mean is 10 and the
              standard error is 2?
         •    Z = (16-10) / 2 = 3.




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                     Another Example
         •    What is the Z value for the mean 16 of 4
              observations, if the average of repeated
              sample of means is 10 and the standard
              deviation of the observations is 2?
         •    Standard deviation of mean =
              2 / 4^0.5 = 2/2 =1
         •    Z value for 16 = (16-10)/1 = 6

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                      Use of Z Values
         •    99.7% of data are between z=3 and z=-3.
         •    Z is the number of standard deviations that
              X is away from the mean.
         •    0.15% of data are below z=-3.
         •    0.15 % of data are above z=3.



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              Use of Z Value (Continued)
         •    95% of data are within z=1.96 and z=-1.96
         •    5% are outside z=1.96 and z=-1.96
         •    2.5% of data are below z=-1.96
         •    2.5% of data are above z=1.96




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                   Confidence Interval
         •    For Normal distributions, the 95% two
              tailed confidence interval corresponds to
              observations where z=1.96 and z=-1.96.




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                           Example
         •    What is the 95% confidence interval for
              mean of 10 and standard deviation of 2?
         •    Lower limit = 10-1.96*2 = 6.08.
         •    Upper limit = 10+1.96*2 =13.92.
         •    At 13.92, Z value is (13.92-10)/2=1.96.
         •    At 6.08 , Z value is (6.08-10) / 2=-1.96.
         •    95% of data fall within these limits.
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              Two Tailed Confidence Interval
         •     What percentage of data are between z=1.96 and
               Z=-1.96. Answer: 95%. Often referred to as
               two-tailed confidence interval.
         •     What percentage of data are below z=1.96?
         •     Answer = 97.5. Often referred to as one tailed-
               confidence interval.
         •     What percentage of data are above Z=-1.96.
               Answer =97.5. Often referred to as one tailed
               confidence interval.
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                          Hypothesis
         •    A statistical hypothesis is a conjecture
              about population parameter.
         •    The null hypothesis is that there is no
              difference between the parameter and a
              value.
         •    The alternative hypothesis states there is a
              specific difference.
                              Experimental data can only reject a
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                                   hypothesis not accept it.
          Possible Outcomes of Hypothesis
                        Test
         There are four possible outcomes:
              1.   We reject a hypothesis that is true.
              2.   We reject a hypothesis that is false.
              3.   We do not reject a hypothesis that is true.
              4.   We do not reject a hypothesis that is false.




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                      Two Types of Error
                            Hypothesis is    Hypothesis is
                               true             false

         We reject          Type one error      Correct
         hypothesis

         We do not reject      Correct       Type two error
         hypothesis

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                          Type 1 Error
         •    The level of significance is the maximum
              probability of type 1 error, symbolized by alpha,
              .
         •    When we base our decision on 95% confidence
              intervals, 5% of the data are ignored at the two
              tails of the distribution. Therefore, there is 5%
              chance that we will reject a hypothesis that is
              true.
         •    Type one error= 5%,  = 0.05.
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                      One-tailed Tests
         •    In a two-tailed test, the hypothesis is
              rejected when the value is above higher
              limit and below the lower limit.
         •    In a one-tailed test that a parameter is
              larger than a particular value, the
              hypothesis is rejected when the value is
              above higher limit.

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              One-tailed Tests (Continued)
         •    When we base our decision on 95%
              confidence intervals, 2.5% of the data are
              ignored at one tail of the distribution.
              Therefore, there is 2.5% chance that we
              will reject a hypothesis that is true.
         •    =0.025.


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              Steps in Testing a Hypothesis
              1. State the null hypothesis.
              2. Identify the alternative hypothesis.
              3. Is this a one tailed or two tailed test?
              4. Decide the critical Z value above or below which the
                 hypothesis is rejected, usually 1.96.
              5. Calculate the Z value corresponding to the
                 observation.
              6. Reject or do not reject the hypothesis by comparing
                 the calculated Z to the critical values.

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                                 Example
         •      The cost of rehabilitation in the industry is
                $25,000, with a standard deviation of
                3000.
         •      In our hospital, the average cost is
                $30,000.
         •      With 95% confidence, would you say that
                our cost is different than the industry?

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              Do this in Excel
                 Steps in Testing Example
                        Hypothesis
              1. The null hypothesis: Our cost is higher or
                 lower than average.
              2. Alternative hypothesis: Our costs are the
                 same as the industry.
              3. This is a two tailed test.
              4. The critical Z is +1.96 or –1.96.
              5. Observed Z = (30000-25000)/3000 = 1.66.
              6. Do not reject the hypothesis.
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               When to Assume Normal
                Distribution for Means
         •    When the population variance is known
              and observations have a Normal
              distribution.
         •    When the population variance is unknown
              and there are more than 30 observations.
         •    Otherwise use t-distribution an
              approximation for Normal distribution.

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                     Use t-distribution
         •    If the values in the population is Normal.
         •    If we have less than 30 observations.
         •    If we have to estimate the standard
              deviation from the sample and variance of
              the population is not known.
         •    The t-distribution is used as an
              approximation for near Normal data.
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                   Calculating t Statistic
         •    t= (observed average – mean) / standard
              deviation of the average.
         •    Critical value of t depends on sample size.
         •    For one tail test of alpha = 0.025 and two
              tailed test of alpha =0.05.
         •    The critical t value for sample size of 10 is
              2.22 and for sample size of 20 is 2.08.
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                  Calculating t Statistic
                      (Continued)
         •    If we are examining sample size of 10,
              95% of data are within t=2.22 and t=-2.22.
         •    If we are examining sample size of 10,
              97.5% of data are below t=2.22.




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                   Testing With t-distribution
              1. State the null hypothesis.
              2. Identify the alternative hypothesis.
              3. Is this a one tailed or two tailed test?
              4. Decide the critical t value above or below which the
                 hypothesis is rejected, the value depends on sample
                 size.
              5. Calculate the t value corresponding to the
                 observation.
              6. Reject or do not reject the hypothesis by comparing
                 the calculated t to the critical values.
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              Example Data




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              Selecting Data Analysis




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              Select Descriptive Statistics




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              Enter Data Range




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              Result
                       Confidence interval is
                       the mean plus or
                       minus the confidence
                       level. If it does not
                       include $30,000, then
                       our hospital has a
                       different cost structure
                       than other hospitals in
                       our database



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