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Testing the Difference Between Two Means Small Dependent Samples_Part 1

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					    CHAPTER 4
HYPOTHESIS TESTING WITH TWO SAMPLES
                                              Introduction



 There are many instances researchers wish to
  compare two sample means.

 Examples:

   The average lifetimes of two different brands of
    batteries might be compared to examine
    whether one brand has longer lifetime than the
    other.
   Two types of materials might be tested to see
    whether there is any difference in tensile
    strength.

 In the comparison of two means, the same basic
  steps for hypothesis testing shown in Chapter 3 are
  used.
4.1
Testing the Difference Between
Two Means: Large Samples


Lesson outcome:

Test the difference between two large
sample means, using the z-test.
               Testing the Difference Between Two Means: Large Samples


Introduction
Suppose Pn Azlyna wants to determine whether there
is a difference in the average score of engineering
students in IQ Test for University A and University B.
In this case Pn Azlyna is not interested in the average
score for both universities.
She is interested in comparing the means of the two
groups.

The hypotheses:
H 0 : 1   2        1  Mean score for University A
H1 : 1   2          2  Mean score for University B
               Testing the Difference Between Two Means: Large Samples



Assumptions for testing the difference between
two means:

 The samples must be independent of each other
  (there can be no relationship between the subjects
  in each sample).

 The population from which the samples were
  obtained must be normally distributed or
  approximately normally distributed, and the
  population standard deviations of the variables
  must be known
                          or

  the sample sizes must be ≥ 30.
                Testing the Difference Between Two Means: Large Samples



 The theory behind testing the difference between
  two means is based on selecting pairs of samples
  and comparing the means of the pairs.

 All possible pairs of samples are taken from
  populations.

 The means for each pair of samples are
  computed and then subtracted.

 After that, the differences are plotted.
               Testing the Difference Between Two Means: Large Samples




                                                  Distribution of

                                                  X1  X 2




                      0

 If both populations have the same mean, then most
  of the differences will be zero or close to zero.

 There will be a few large differences due to chance
  alone, some positive and others negative.

 If the differences are plotted, the curve will be
  shaped like the normal distribution and have mean
  zero.
               Testing the Difference Between Two Means: Large Samples



Formula for the z-test for comparing two
means from independent populations


         ( X 1  X 2 )  ( 1   2 )
    z
                   12       2
                              2
                         
                   n1        n2
                                                               4.1


Based on the general format

             (observed value)  (expected value)
Test value 
                       standard error
              Testing the Difference Between Two Means: Large Samples



 If  1 and  2 are not known, use s1 and s2
       2       2                     2      2




 But both sample sizes must be n1 ≥ 30 and n2 ≥ 30




        ( X 1  X 2 )  ( 1   2 )
   z
                     2         2
                    s   s
                     1
                              2
                    n1 n2
                                                              4.2
                           Testing the Difference Between Two Means: Large Samples




 The tests can be one-tailed or two-tailed using the
  following hypotheses:


                                     H0 : μ1 = μ2   H :μ -μ =0
   Two-tailed test                                OR 0 1 2
                                     H1 : μ1 ≠ μ2   H1 : μ1 - μ2 ≠ 0

                                  H0 : μ1 ≤ μ2    H0 : μ1 - μ2 ≤ 0
   One-tailed test



                     Right-tailed              OR
                                  H1 : μ1 > μ2    H1 : μ1 - μ2 > 0

                                     H0 : μ1 ≥ μ2    H0 : μ1 - μ2 ≥ 0
                     Left-tailed                  OR
                                     H1 : μ1 < μ2    H1 : μ1 - μ2 < 0


                                   We have one tail
                                   or two tails ???
                    Testing the Difference Between Two Means: Large Samples

Hypothesis testing situations in the comparison of means



 Sample 1            Sample 2          Sample 1            Sample 2
     X1                  X2                 X1                 X2




          Population                  Population         Population
            μ1 = μ2                      μ1                 μ2



  Difference is not significant          Difference is significant

  Do not reject   H 0 : 1   2         Reject   H 0 : 1   2
  X 1  X 2 is not significant           X 1  X 2 is significant
                Testing the Difference Between Two Means: Large Samples



Example 4.1
A researcher hypothesizes that the average
number of sports that private colleges offer for
males is greater than the average number of
sports offer for females. A sample of the number of
sports offered by colleges is collected and the
results are given below:

For the males     :   n1  50 ,     X 1  8.6, s1  3.3
For the females :     n2  50 ,      X 2  7.9, s2  3.3

At α = 0.10, is there enough evidence to support
the claim.
                Testing the Difference Between Two Means: Large Samples




 Sometimes, we are interested in testing a specific
  difference in means other than zero.

 Example: For Example 4.1, the researcher might
  hypothesize that the average number of sports that
  private colleges offer for males is more than three
  compared to the average number of sports offer for
  females. In this case the hypotheses are:

                 H 0 : 1   2  3
                 H1 : 1   2  3
                      Testing the Difference Between Two Means: Large Samples


Confidence intervals for the difference
between two means: large samples
When one is hypothesizing a difference of 0, if the
☻ confidence interval contains 0 → H0 is not rejected.

☻ confidence interval does not contain 0 → H0 rejected.


                               12       2
                                          2
  ( X1  X 2 )  z / 2                       1   2 
                               n1        n2
                               12       2
                                          2
  ( X1  X 2 )  z / 2              
                               n1        n2                                 4.3

Note: The relationship presented here is valid for two-tailed tests only.
4.2
Testing the Difference Between
Two Variances


Lesson outcome:

Test the difference between two variances
or standard deviations.
                       Testing the Difference Between Two Variances


Introduction

 In addition to comparing two means, a researcher
  may be interested in comparing two variances or
  standard deviations.

 Example: Is the variation in the temperatures for a
  certain month for two cities different?

 For the comparison of two variances or standard
  deviations, an F-test is used.

 The F-test should not be confused with the Chi-
  square test, which compares a single sample
  variance to a specific population variance, as
  shown in Chapter 3.
                         Testing the Difference Between Two Variances


Characteristics of the F Distribution

 The values of F cannot be negative, because
  variances are always positive or zero.

 The distribution is positively skewed.

 The mean value of F is approximately equal to 1.

 The F distribution is a family of curves based on
  the degrees of freedom of the variance of the
  numerator and the degrees of freedom of the
  variance of the denominator.
                       Testing the Difference Between Two Variances



Assumptions for testing the difference
between two variances

 The populations from which the samples were
  obtained must be normally distributed.
  (The test should not be used when the distribution
  depart from normality.)

 The samples must be independent of each other,
                      Testing the Difference Between Two Variances


Formula for the F-test for testing the difference
between two variances

               2
              s1
            F 2
              s2                                 4.4


   s12  s2
           2


 F test has two terms for degrees of freedom (DOF):
    ☻DOF of the numerator,       v1  n1  1
    ☻DOF of the denominator,     v2  n2  1

n1 is the sample size from which the larger variance
    was obtained
                         Testing the Difference Between Two Variances


Notes for the use of the F-test

 For a two-tailed test, the α value must be divided by
  2 and the critical value placed on the right side of
  the F-curve.
  Note: When one conducting a two-tailed test, α is
  split and even though there are two values, only the
  right tail is used. This is because the F test value is
  always ≥ 1

 If the standard deviations instead of the variances
  are given in the problem, they must be squared for
  the formula for the F-test.

 When the DOF cannot be found from TABLE 9, the
  closest value on the smaller side should be used.
  Note: This guideline keeps type I error ≤ α
                          Testing the Difference Between Two Variances



Example 4.2

A medical researcher wishes to see whether the
variance of the heart rates (in beats per minute) of
smokers is different from the variance of heart rates
of people who do not smoke. Two samples are
selected and data are shown below. Using α = 0.05,
is there enough evidence to support the claim?

              Smokers            Nonsmokers

               n1  26              n2  18
               s12  36             s2  10
                                     2
                             Testing the Difference Between Two Variances


Note
 It is not necessary to place the larger variance in the
  numerator when one performing the F-test (specifically
  in left-tailed test)

 Critical values for left-tailed hypotheses tests can be
  found by interchanging the DOF and taking the
  reciprocal (salingan) of the value found in TABLE 9.

 Caution:     When performing the F-test, the data can
  run contrary to the hypotheses on rare occasions.

 Example:      If the hypotheses are
                H 0 :  12   2
                               2

                                       but      s12  s2
                                                       2
                H1 :  12   2
                              2


  then F-test should not be performed and one would
  not reject H0.
                       Testing the Difference Between Two Variances



Example 4.3

A mathematics professor claims that the variance on
placement tests when the students use graphing
calculators will be smaller than the variance on
placement tests when the students use scientific
calculator. A randomly selected group of 50 students
who used graphing calculators had a variance of 32,
and a randomly selected group of 40 students who
used scientific calculators had a variance of 37. Is
the professor correct, using α = 0.05?
                        Testing the Difference Between Two Variances



Example 4.4

Find the p-value interval for each F-test value.


(a)   F  2.97 , v1  9, v2  14, right-tailed test
(b)   F  3.32, v1  6, v2  12, two-tailed test
(c)   F  1.16, v1  6, v2  12, left-tailed test