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
Are 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.

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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

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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
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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.
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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

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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

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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.

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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
• (Note this is F.025,30,29 since the table does
  not give the value for F.025, 31,29)
• Calculation sW2/ sM2 (13.92)2/(11.79)2 1.39
• Since 1.39 lt 2.09, Cannot conclude unequal
  variances.
• Do Equal Variance t-test with 3230-260 degrees
  of freedom.

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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.

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Example 195 Confidence Interval
95 Confidence Interval
2 6.57 -4.57 ?? 8.57
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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.

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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.

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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.
20
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.

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Example 195 Confidence Interval
95 Confidence Interval
4666 2238.47 2427.53 ?? 6904.47
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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.

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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)

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Excel For Example 1 F-Test
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Example 1 F-Test (Contd)
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Example 1 F-Test (Contd)
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Example 1 F-Test (Contd)
High p-value (.371671) Cannot conclude Unequal
Variances Use Equal Variance t-test
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Example 1 t-Test
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Example 1 t-Test (Contd)
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Example 1 t-test (Contd)
High p-value for 1-tail test! Cannot conclude
average womens score gt average mens score
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Example 1 95 Confidence Interval

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Excel For Example 2 F-Test
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Example 2 F-Test (Contd)
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Example 2 F-Test (Contd)
Low p-value (.000352) Can conclude Unequal
Variances Use Unequal Variance t-test
35
Example 2 t-Test
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Example 2 t-Test (Contd)
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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.
38
Example 2 95 Confidence Interval
39
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