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Two Population Means

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Can we conclude that the Lakers average attendance is more than 2000 more than ... Use Lakers (Column B) for Variable Range 1. Use Clippers (Column D) for ... – PowerPoint PPT presentation

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Title: Two Population Means


1
  • Two Population Means
  • Hypothesis Testing and Confidence Intervals
  • With Unknown
  • Standard Deviations

2
The Problem
  • ?1 or ?2 are unknown
  • ?1 and ?2 are not known (the usual case)
  • OBJECTIVES
  • Test whether ?1 gt ?2 (by a certain amount)
  • or whether ?1 ? ?2
  • Determine a confidence interval for the
    difference in the means ?1 - ?2

3
KEY ASSUMPTIONS
  • Sampling is done from two populations.
  • Population 1 has mean µ1 and variance s12.
  • Population 2 has mean µ2 and variance s22.
  • A sample of size n1 will be taken from population
    1.
  • A sample of size n2 will be taken from population
    2.
  • Sampling is random and both samples are drawn
    independently.
  • Either the sample sizes will be large or the
    populations are assumed to be normally
    distribution.

4
Distribution of ?X1 - ?X2
  • Since X1 and X2 are both assumed to be normal, or
    the sample sizes, n1 and n2 are assumed to be
    large, then because ?1 and ?2 are unknown, the
    random variable ?X1 -?X2 has a
  • Distribution -- t
  • Mean ?1 - ?2
  • Standard deviation that depends on whether or not
    the standard deviations of X1 and X2 (although
    unknown) can be assumed to be equal
  • Degrees of freedom that also depends on whether
    or not the standard deviations of X1 and X2 can
    be assumed to be equal

5
Appropriate Standard Deviation For ?X1 -?X2 When
ss Are Known
  • Recall the appropriate standard deviation for ?X1
    - ?X2 is
  • Now if ?1 ?2 we can simply call it ? and write
    it as
  • So if the standard deviations are unknown, we
    need an estimate for the common variance, ?2.

6
Estimating ?2 Degrees of Freedom
  • If we can assume that the populations have equal
    variances, then the variance of ?X1 - ?X2 is the
    weighted average of s12 and s22, weighted by
  • DEGREES OF FREEDOM
  • There are n1- 1 degrees of freedom from the first
    sample and n2-1 degrees of freedom from the
    second sample, so
  • Total Degrees of Freedom for the hypothesis test
    or confidence interval (n1 -1) (n2 -1) n1
    n2 -2

7
The Appropriate Standard DeviationFor ?X1 - ?X2
When Are ss Unknown, but Can Be Assumed to Be
Equal
  • The best estimate for ?2 then is the pooled
    variance, sp2
  • Thus the best estimates for the variance and
    standard deviation of ?X1 - ?X2 are

8
t-Statistic and t-Confidence Interval Assuming
Equal Variances

Confidence Interval
Degrees of Freedom n1 n2 -2
9
The Appropriate Standard DeviationFor ?X1 - ?X2
When Are ss Unknown, And Cannot Be Assumed to Be
Equal
  • If we cannot assume that the populations have
    equal variances, then the best estimate for ?12
    is s12 and the best estimate for ?22 is s22.
  • Thus the best estimates for the variance and
    standard deviation of ?X1 - ?X2 are

10
t-Statistic and t-Confidence Interval Assuming
Unequal Variances

Confidence Interval
Total Degrees of Freedom
Round the resulting value.
11
?s are knownz-distribution
  • Standard Error

12
?s are unknownt-distribution
13
Testing whether the Variances Can Be Assumed to
Be Equal
  • The following hypothesis test tests whether or
    not equal variances can be assumed
  • H0 s12/s22 1 (They are equal)
  • HA s12/s22 ? 1 (They are different)
  • This is an F-test!
  • If the larger of s12 and s22 is put in the
    numerator, then the test is
  • Reject H0 if F s12/s22 gt Fa/2, DF1, DF2

14
Hypothesis Test/Confidence Interval Approach With
Unknown ?s
  • Take a sample of size n1 from population 1
  • Calculate ?x1 and s12
  • Take a sample of size n2 from population 2
  • Calculate ?x2 and s22
  • Perform an F-test to determine if the variances
    can be assumed to be equal
  • Perform the Appropriate Hypothesis Test or
    Construct the Appropriate Confidence Interval

15
Example 1
  • Based on the following two random samples,
  • Can we conclude that women on the average score
    better than men on civil service tests?
  • Construct a 95 for the difference in average
    scores between women and men on civil service
    tests.
  • Because the sample sizes are large, we do not
    have to assume that test scores have a normal
    distribution to perform our analyses.

16
Example 1 F-test
  • Do an F-test to determine if variances can be
    assumed to be equal.
  • H0 ?W2/?M2 1 (Equal Variances)
  • HA ?W2/?M2 ? 1 (Unequal Variances)
  • Select a .05.
  • Reject H0 (Accept HA) if Larger s2/Smaller s2
    gt F.025,DF(Larger s2),DF(Smaller s2)
    F.025,31,29 2.09
  • Calculation sW2/ sM2 (13.92)2/(11.79)2 1.39
  • Since 1.39 lt 2.09, Cannot conclude unequal
    variances.

17
Example 1 The Equal Variance t-Test
Example 1 The Equal Variance t-Test
  • H0 ?W - ?M 0
  • HA ?W - ?M gt 0
  • Select a .05.
  • Reject H0 (Accept HA) if t gt t.05,60 1.658
  • Since .608 lt 1.658, we cannot conclude that
  • women average better than men on the tests.

18
Example 195 Confidence Interval
95 Confidence Interval
2 6.57 -4.57 ?? 8.57
19
Example 2
  • Based on the following random samples of
    basketball attendances at the Staples Center,
  • Can we conclude that the Lakers average
    attendance is more than 2000 more than the
    Clippers average attendance at the Staples
    Center?
  • Construct a 95 for the difference in average
    attendance between Lakers and Clippers games at
    the Staples Center.
  • Since sample sizes are small, we must assume that
    attendance at Lakers and Clipper games have
    normal distributions to perform the analyses.

20
Example 2 F-test
  • Do an F-test to determine if variances can be
    assumed to be equal.
  • H0 ?C2/?L2 1 (Equal Variances)
  • HA ?C2/?L2 ? 1 (Unequal Variances)
  • Note Clipper variance is the larger sample
    variance
  • Choose a .05.
  • Reject H0 (Accept HA) if Larger s2/Smaller s2 gt
    F.025,DF(Larger variance),DF(Smaller variance)
    F.025,10,12 3.37
  • Calculation sC2/ sL2 (3276.73)2/(1014.97)2
    10.42
  • Since 10.42 gt 3.37, Can conclude unequal
    variances.
  • Do Unequal Variance t-test.

21
Degrees of Freedom for the Unequal Variance t-Test
  • The degrees of freedom for this test is given by

11.626
This rounded to 12 degrees of freedom.
22
Example 2 the t-Test
  • Proceed to the hypothesis test for the difference
    in means with unequal variances
  • H0 ?L - ?C 2000
  • HA ?L - ?C gt 2000
  • Select a .05.
  • Reject H0 (Accept HA) if t gt t.05,12 1.782
  • Since t 2.595 gt 1.782, we can conclude that the
    Lakers average more than 2000 per game more than
    the Clippers at the Staples Center.

23
Example 295 Confidence Interval
95 Confidence Interval
4666 2238.47 2427.53 ?? 6904.47
24
Excel Approach
  • F-test, t-test Assuming Equal Variances, t-test
    Assuming Unequal Variances are all found in Data
    Analysis.
  • Excel only performs a one-tail F-test.
  • Multiply this 1-tail p-value by 2 to get the
    p-value for the 2-tail F-test.
  • Formulas must be entered for the LCL and UCL of
    the confidence intervals.
  • All values for these formulas can be found in the
    Equal or Unequal Variance t-test Output.

25
Inputting/Interpreting Results From Hypotheses
Tests
  • Express H0 and HA so that the number on the right
    side is positive (or 0)
  • The p-value returned for the two-tailed test will
    always be correct.
  • The p-value returned for the one-tail test is
    usually correct. It is correct if
  • HA is a gt test and the t-statistic is positive
  • This is the usual case
  • If t lt 0, the true p-value is 1 (p-value
    printed by Excel)
  • HA is a lt test and the t-statistic is negative
  • This is the usual case
  • If tgt0, the true p-value is 1 (p-value printed
    by Excel)

26
Excel For Example 1 F-Test
27
Example 1 F-Test (Contd)
28
Example 1 F-Test (Contd)
29
Example 1 F-Test (Contd)
High p-value (.371671) Cannot conclude Unequal
Variances Use Equal Variance t-test
30
Example 1 t-Test
31
Example 1 t-Test (Contd)
32
Example 1 t-test (Contd)
High p-value for 1-tail test! Cannot conclude
average womens score gt average mens score
33
Example 1 95 Confidence Interval
34
Excel For Example 2 F-Test
35
Example 2 F-Test (Contd)
36
Example 2 F-Test (Contd)
Low p-value (.000352) Can conclude Unequal
Variances Use Unequal Variance t-test
37
Example 2 t-Test
38
Example 2 t-Test (Contd)
39
Example 2 t-test (Contd)
Low p-value for 1-tail test (compared to a
.05)! Can conclude the Lakers average more than
2000 more people per game than the Clippers.
40
Example 2 95 Confidence Interval
41
Review
  • Standard Errors and Degrees of Freedom when
  • Variances are assumed equal
  • Variances are not assumed equal
  • F-statistic to determine if variances differ
  • t-statistic and confidence interval when
  • Variances are assumed equal
  • Variances are not assumed equal
  • Hypothesis Tests/ Confidence Intervals for
    Differences in Means (Assuming Equal or Unequal
    Variances)
  • By hand
  • By Excel
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