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Statistics- Tests of Hypotheses -2

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					                                        1
          Tests of Hypotheses:
Chapter
          Large Samples


                            Rejection
                            region
               Acceptance
               region
            GOALS               2
TO DEFINE HYPOTHESES AND
 HYPOTHESIS TESTING.
TO DESCRIBE THE HYPOTHESIS TESTING
 PROCEDURE.
TO DISTINGUISH BETWEEN ONE-TAILED
 AND TWO-TAILED TEST OF HYPOTHESIS.
TO CONDUCT A TEST FOR THE
 POPULATION MEAN OR PROPORTION.
           GOALS               3
TO CONDUCT A TEST FOR THE
 DIFFERENCE BETWEEN TWO
 POPULATION MEANS OR PROPORTIONS.
TO DESCRIBE STATISTICAL ERRORS
 ASSOCIATED WITH HYPOTHESIS
 TESTING.
       WHAT IS A HYPOTHESIS?                        4
 Hypothesis: A statement about the value of
  a population parameter developed for the
  purpose of testing.
 Examples of hypotheses, or statements,
  made about a population parameter are:
  » The mean monthly income from all sources for
    systems analysts is $3,625.
  » Twenty percent of all juvenile offenders ultimately
    are caught and sentenced to prison.
  WHAT IS HYPOTHESIS TESTING?           5
 Hypothesis testing: A procedure, based on
 sample evidence and probability theory,
 used to determine whether the hypothesis
 is a reasonable statement and should not
 be rejected, or is unreasonable and should
 be rejected.
 Following is a five-step procedure for
 testing a hypothesis.
STEP 1       State null H0 and alternative      6
                    hypotheses H1

STEP 2     Select a level of significance


STEP 3     Identify the test statistic

STEP 4        Formulate a decision rule

STEP 5   Take a sample , arrive at a decision

                               Reject H0 and
         Do not reject H0      accept H1
 Null Hypothesis H0: A statement about
  the value of a population parameter.
                                          7
 Alternative Hypothesis H1: A statement that
  is accepted if the sample data provide
  evidence that the null hypothesis is false.
 Level of Significance: The probability of
  rejecting the null hypothesis when it is
  actually true.
 Type I Error: Rejecting the null hypothesis,
  H0, when it is actually true.
 Type II Error: Accepting the null
  hypothesis, H0, when it is actually false.
                                              8
                          Researcher
Null hypothesis      Accepts H0   Rejects H0


                     Correct      Type I
If H0 is true and    Decision     error


                      Type II      Correct
If H0 is false and
                      error        Decision
 Test statistic: A value, determined from
  sample information, used to determine
                                             9
  whether or not to reject the null hypothesis.
 Critical value: The dividing point between
  the region where the null hypothesis is
  rejected and the region where it is not
  rejected.
ONE-TAILED TESTS OF SIGNIFICANCE            10
 A test is one-tailed when the alternate
  hypothesis, H1, states a direction such as:
 H0: The mean income of the females is less
  than or equal to the mean income of the
  males.
 H1: The mean income of the females is
  greater than the mean income of the males.
 The region of rejection in this case is to the
  right (upper) tail of the curve. An example
  is shown next:
Sampling Distribution for the Statistic Z for a
 One-Tailed Test, 0.05 Level of Significance      11

                                      Critical
                                      value
                                      1.645 = Z

                                         Region of
                                         rejection
                   0.95 Probability
                                        0.05
                                                     z
TWO-TAILED TESTS OF SIGNIFICANCE              12
 A test is two-tailed when no direction is
  specified in the alternate hypothesis H1,
  such as:
 H0: The mean income of the females is
  equal to the mean income of the males.
 H1: The mean income of the females is not
  equal to the mean income of the males.
 The region of rejection in this case is
  divided equally into the two tails of the
  curve. An example is shown next:
Sampling Distribution for the Statistic Z for a
 Two-Tailed Test, 0.05 Level of Significance         13

              Critical                Critical
              value                   value
                          Do not
              -1.96 = Z               1.96 = Z
                          reject H0
                                         Region of
       Region of                         rejection
       rejection
                          0.95              0.025
      0.025

                                                     z
 TESTING FOR THE POPULATION
MEAN: LARGE SAMPLE, POPULATION      14
  STANDARD DEVIATION KNOWN
 The test statistic is given by:




             z X
                / n
               EXAMPLE                         15
 The processors of Mets Catsup indicate on the
 label that the bottle contains 16 ounces of catsup.
 Mets’ Quality Control Department is responsible
 for monitoring the amount included in the bottle.
 A sample of 36 bottles is selected hourly and the
 contents weighed. Last hour a sample of 36
 bottles had a mean weight of 16.12 ounces with a
 standard deviation of 0.5 ounces. At the 0.05
 significance level can we conclude that the
 process is out of control? (i.e. not meeting weight
 goals)
         EXAMPLE (continued)                   16
 Step 1: State the null and the alternative
  hypotheses.
 H0:  = 16       H1:  16
 Step 2: State the decision rule.
 H0 is rejected if z < -1.96 or z > 1.96.(2-side)
 Step 3: Compute the value of the test
  statistic.
 z= [16.12 - 16]/[0.05/36] = 1.44.
 Step 4: What is the decision on H0?
 H0 is not rejected, because 1.44 is less than
  the critical value of 1.96.
Sampling Distribution for the Statistic Z for a
 Two-Tailed Test, 0.05 Level of Significance              17

              Critical                    Critical
              value                       value
                          Do not
              -1.96 = Z                   1.96 = Z
                          reject H0
                                              Region of
       Region of                 Test Stat.   rejection
       rejection                    1.44
                          0.95                   0.025
      0.025

                                                          z
  TESTING FOR THE POPULATION
MEAN: LARGE SAMPLE, POPULATION            18
 STANDARD DEVIATION UNKNOWN
 Here is unknown, so we estimate it with
the sample standard deviation s.
As long as the sample size n 30, z can be
approximated with


              z X
                 s/ n
              EXAMPLE                       19
 The Thompson’s Discount Store chain issues
 its own credit card. The credit manager wants
 to find out if the mean monthly unpaid balance
 is more than $400. The level of significance is
 set at 0.05. A random check of 172 unpaid
 balances revealed the sample mean to be $407
 and the sample standard deviation to be $38.
 Should the credit manager conclude that the
 population mean is greater than $400, or is it
 reasonable to assume that the difference of $7
 ($407 - $400) is due to chance?
        EXAMPLE (continued)                    20
 Step 1: State the null and the alternative
  hypotheses.
 H0:   400       H1: > 400
 Step 2: State the decision rule.
 H0 is rejected if z > 1.645. (one-sided)
 Step 3: Compute the value of the test statistic.
 z = [407 - 400]/[38/172] = 2.42.
 Step 4: What is the decision on H0?
 H0 is rejected. The manager can conclude that
  mean unpaid balance is greater than $400.
                        21



             Computed
             z = 2.42


Rejection
region


   0        1.645       z
     HYPOTHESIS TESTING: TWO
        POPULATION MEANS
                                                22
Assume the parameters for the two populations
are 1,2,1,and 2.
Case I: When 1,2are known, the test statistic

        is:

                        X1  X 2
                z
                            1
                             2
                                     2
                                      2
                                 
                            n1       n2
    HYPOTHESIS TESTING: TWO
       POPULATION MEANS                   23
 Case II: When 1,2are unknown but the
 sample sizes n1 and n2 are greater than or
 equal to 30, the test statistic is :

              X1  X 2
        z       2        2
                s1       s2
                     
                n1       n2
               EXAMPLE
                                             24
 A study was conducted to compare the mean
 years of service for those retiring in 1975 with
 those retiring last year Kentucky
 Manufacturing Co. The following sample data
 was obtained. At the 0.01 significance level
 can we conclude that the workers retiring last
 year had more service?
        EXAMPLE (continued)                 25
 State the null and the alternative
  hypotheses:
 Let population 2 refer to those that retired
  last year.     H0:2  1     H1:2 > 1
 State the decision rule.
 Reject H0 if z > 2.33. (one-sided)
 Compute the value of the test statistic.
    z  304  256  680
           .     .   .
         362  2.92
          .
         45     40
        EXAMPLE (continued)                26
 What is the decision on the null
  hypothesis? Interpret the results.
 Since z = 6.80 > 2.33, H0 is rejected. Those
  retiring last year had more years of service.
 TESTS CONCERNING PROPORTION               27
 Proportion: A fraction or percentage that
 indicates the part of the population or
 sample having a particular trait of interest.

  Sample proportion is denoted by p where

  p Number of successes in the sample
            Number sampled
  TEST STATISTIC FOR TESTING A   28
SINGLE POPULATION PROPORTION



z    p p
     p(1 p)
        n
p  population proportion
p  sample proportion
               EXAMPLE                      29
 In the past 15% of the mail order
 solicitations for a certain charity resulted in
 a financial contribution. A new solicitation
 letter has been drafted. Will this new letter
 increase the solicitation rate? The new
 letter is sent to a sample of 200 people and
 45 responded with a contribution. At the
 0.05 significance level can it be concluded
 that the new letter is more effective?
        EXAMPLE (continued)                  30
 State the null and the alternative
  hypotheses:
 H0:p  0.15    H1:p > 0.15.
 State the decision rule.
 H0 is rejected if z > 1.645. (one-sided)
 Compute the value of the test statistic.
          45  015
                .
     z  200         2.97
           .     .
         (015)(085)
             200
         EXAMPLE (continued)                31
 What is the decision on the null
  hypothesis? Interpret the results.
 Since z = 2.97 > 1.645, H0 is rejected. The
  new letter is more effective.
A TEST INVOLVING THE DIFFERENCE               32
    BETWEEN TWO POPULATION
          PROPORTIONS
 The test statistic in this case is

                     p1  p2
     z
            pc (1  pc )       pc (1  pc )
                           
                n1                 n2

n1 is the sample size from population 1.
n2 is the sample size from population 2.
 TWO PROPORTIONS (continued)            33
pc is the weighted mean of the two sample
   proportions, computed by:
      Total number of successes X 1  X 2
 pc                           =
       Total number in samples   n1  n2
X1 is the number of successes in n1.
X2 is the number of successes in n2.
              EXAMPLE                    34
 Are unmarried workers more likely to be
  absent from work than married workers? A
  sample of 250 married workers showed 22
  missed more than 5 days last year for any
  reason. A sample of 300 unmarried
  workers showed 35 missed more than 5
  days. Use the 0.05 significance level.
 State the null and alternative hypotheses.
 H0:p2  p1    H1:p2 > p1 where subscript 2
  refers to the population of unmarried
  workers.
        EXAMPLE (continued)                  35
 State the decision rule.
 H0 is rejected if z > 1.645. (one-sided)
 Compute the value of the test statistic.

   p  22  35  01036
                  .
      250 300
    c




   Z           01167  00880
                  .      .               110
                                           .
       01036(1 01036)  01036(1 01036)
        .         .        .       .
              300             250
        EXAMPLE (continued)                 36
 What is the decision on the null
  hypothesis?
 H0 is not rejected. There is no difference in
  the proportion of married and unmarried
  workers missing more than 5 days of work.
                              37
       For Final Exam

Chapters 12 and 13:
 Regression, Inference, and
 Model Building/ ANOVA

          James S. Hawkes

				
DOCUMENT INFO
Description: Prof Rushen's Notes for MBA/ BBA students