confidence interval hypothesis by MA826S

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									               Estimation and Hypothesis
                  (Hadyana Sukandar)
                    Lesson Objectives


1. Know what is parameter estimation
2. Understand hypothesis testing & the “types of errors” in
   decision making.
3. Learn how to use test statistics to examine hypothesis
   about population mean, proportion
         Statistic           Parameter

Mean:         X        estimates       ____
Standard
deviation:    s        estimates       ____

Proportion:   p        estimates       ____
                                   from entire
         from sample
                                   population
Population     Point estimate Interval estimate
                                     I am 95%
                  Mean           confident that 
 Mean, , is                     is between 40 &
                  X = 50
 unknown                                 60

 Sample
CONFIDENCE INTERVAL
Definitions :
1. Estimator :
   a formula or process for using sample data to
   estimate a population parameter
2. Estimate :
   a specific value or range of values used to
   approximate some population parameter
3. Point Estimate :
   a single value (or point) used to approximate a
   population parameter
  The sample mean x is the best point estimate of
   the population mean µ
4. Confidence Interval (or Interval Estimate) :
   a range (or an interval) of values used to
   estimate the true value of the population
   parameter.

   Lower # < population parameter < Upper #

     As an example :
     Lower # < µ< Upper #
5. Degree of confidence (level of confidence or
   confidence coefficient) :
   The probability 1 -  (often expressed as the
   equivalent percentage value) that is the relative
   frequency of times the confidence interval actually
   does contain the population parameter, assuming that
   the estimation process is repeated a large number of
   times.

  Usually 95 %, (= 5%) or 99 % (=1 %).
Interpreting a Confidence Interval for  (for example :
Systolic Blood pressure of medical student) :
               100 <  < 140

We are 95 % confident that the interval from 100 to 140
actually does contain the true value of . This means
that if we were to select many different samples of
size 100 and construct the confidence interval, 95 % of
them would actually contain the value of the population
mean .

     PARAMETER = STATISTIC  ITS ERROR
               Standard Error

                                        S
Quantitative Variable   SE (Mean) =    n

                                      p(1-p)
Qualitative Variable    SE (p) =        n
      Confidence Interval


α/2                                         α/2
                   1-α
                                                   _
                                                   X
              SE            SE           Z-axis

               95% Samples

        X - 1.96 SE       X + 1.96 SE
      Confidence Interval


α/2                                       α/2
                  1-α

             SE           SE           Z-axis   p

              95% Samples

        p - 1.96 SE       p + 1.96 SE
                    Interpretation of
                           CI



  Probabilistic                         Practical



In repeated sampling 100(1-)%    We are 100(1-)% confident that
 of all intervals around sample
                                  the single computed CI contains
    means will in the long run
             include                           
     Example (Sample size≥30)

An epidemiologist studied the blood glucose
level of a random sample of 100 patients. The
mean was 170, with a SD of 10.
                            = X + Z SE
SE = s/√n = 10/10 = 1
                                            95
Then CI:                                    %


 = 170 + 1.96  1 168.04   ≤ 171.96
           Example (Proportion)
In a survey of 140 asthmatics, 35% had allergy to
  house dust. Construct the 95% CI for the
  population proportion.
  = p + Z √ p(1-p)/n     SE= √ 0.35(1-0.35)/140 = 0.04


0.35 – 1.96  0.04   0.35 + 1.96  0.04
               0.27   ≤ 0.43
              27%    43%
            HYPOTHESIS


               What is Hypothesis ?

Definition :

Hypothesis in statistics, is a claim or statement

about a property of population.
Components of formal hypothesis test :
- Null Hypothesis : H0
  - Statement about value of population parameter
  - Must contain condition of equality : = ; ; or 
  - Test the null Hypothesis directly
  - Recect H0 or fail to reject H0
Alternative Hypothesis
 - Must be true if H0 is false
 - ≠; < ; >
 - Opposite of null

Note : If you are conducting a study and want to
 use a hypothesis test to support your claim, the
 claim must be worded so that it becomes the
 alternative hypothesis.
LEVEL OF SIGNIFICANCE, 

The probability that the test statistic will fall in
the critical region when the null hypothesis is
actually true. Common choices are 0.05; 0.01
and 0.10.
TYPE ERROR : TYPE I AND TYPE I

Type I error : - the mistake of rejecting the null
                 hypothesis when it is true.
               -  (alpha) is used to represent the
                 probability of a type I error.

Example : Rejecting a claim that the mean systolic
 blood pressure is 110 mmHg when the mean really
 does equal 110 mmHg.
Type II error : - the mistake of failing to reject the null
                  hypothesis when it is false.
                -  (beta) is used to represent the
                  probability of a type II error.



Example : Failling to reject the claim that the mean
systolic blood pressure is 110 mmHg when the mean
really different from 110 mmHg.
   Table Type I and Type II Errors

                          True State of Nature

           The null hypothesis is    The null hypothesis is
           true                      false
Decision : Type I error              Correct decision
           (rejecting a true null
           hypothesis)
                    
           Correct decision          Type II error
                                     (rejecting a falls null
                                     hypothesis)
                                             
p value Test

- Probability of Obtaining a Test Statistic More
  Extreme ( or  ) than Actual Sample Value
  Given H0 Is True .
- Called Observed Level of Significance
- Used to Make Rejection Decision
  * If p value  , Do Not Reject H0
  * If p value < , Reject H0
Hypothesis Testing: Steps

Test the Assumption that the true mean SBP of
participants is 120 mmHg.

* State H0                H0 : µ = 120
* State H1               H1 : µ  120
* Choose              = 0.05
* Choose n            n = 100
* Choose Test:        Z, t, X2 Test (or p Value)
* Compute Test Statistic (or compute P value)
* Search for Critical Value
* Make Statistical Decision rule
* Express Decision
     Example : One sample-mean Test

- Assumptions
- Population is normally
   distributed
- State the null and
   alternative hypotheses
   H0: µ = µ0
   H1: µ  µ0
      - t test statistic :
                              sample mean  null value x   0
                           t                         
                                   standard error       s
                                                           n

								
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