Comparing Two Groups Means or Proportions

1
Comparing Two Groups Means or Proportions

• Independent Samples t-tests

2
Review

• Confidence Interval for a Mean
• Slap a sampling distribution over a sample mean
  to determine a range in which the population mean
  has a particular probability of beingsuch as 95
  CI.
• If our sample is one of the middle 95, we know
  that the mean of the population is within the CI.

Significance Test for a Mean Slap a sampling
distribution over a guess of the population mean
to determine if the sample has a very low
probability of having come from a population
where the guess is truesuch as a-level .05. If
our sample mean is in the outer 5, we know to
reject the guess, our sample has a low chance of
having come from a population with the mean we
guessed.
Y-bar?
Y-bar?
µ?
2.5
2.5
2.5
2.5
-1.96z
1.96z
Y-bar 95 CI Y-bar /- 1.96 (s.e.)
-1.96z
1.96z
µoguess z or t (Y-bar - µo)/ s.e.
20 21 22 23 24 X H 27 28 29 30
sampling distribution the way statistics for
samples of a certain size would stack up or be
distributed after all possible samples are
collected
3
Review

• Lets collect some data on educational
  aspirations and produce a 95 confidence interval
  to tell us where the population parameter likely
  falls and then lets do a test of significance
  where we guess that average aspiration will be 16
  years.
• I collected a sample of 625 kids who reported
  their educational aspirations where 12 high
  school, 16 equals 4 years of college and so
  forth. The average for the sample was 15 years
  with a standard deviation of 2 years.
• 95 confidence interval 95 CI Sample Mean /-
  z s.e.
• Find the standard error of the sampling
  distribution
• s / ?n 2/v625 2/25 0.08
• Build the width of the Interval. 95 corresponds
  with a z of /- 1.96.
• /- z s.e 1.96 0.08 0.157
• Insert the mean to build the interval
• 95 CI Sample Mean /- z s.e 15 /-
  0.157
• The interval 14.84 to 15.16
• We are 95 confident that the population mean
  falls between these values. (What does this say
  about my guess???)

4
Review

• Lets collect some data on educational
  aspirations and produce a 95 confidence interval
  to tell us where the population parameter likely
  falls and then lets do a test of significance
  where we guess that average aspiration will be 16
  years.
• I collected a sample of 625 kids who reported
  their educational aspirations where 12 high
  school, 16 equals 4 years of college and so
  forth. The average for the sample was 15 years
  with a standard deviation of 2 years.
• Significance Test z or t (Y-bar - µo)/ s.e.
• Decide ?-level (? .05) and nature of test
  (two-tailed)
• Set critical z or t (/- 1.96)
• Make guess or null hypothesis,
• Ho ? 16
• Ha ? ? 16
• Collect and analyze data
• Calculate Z or t z/t Y-bar - ?o
  (s.e. s/vn 2/v625 2/25 .08)
• 
  s.e.
• z/t (15 16)/.08 -1/.08 -12.5
• Make a decision about the null hypothesis (reject
  the null -12.5 lt -1.96)
• Find the P-value (look up 12.5 in z or t table).
  P lt .0001
• It is extremely unlikely that our sample came
  from a population where the mean is 16.

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Other Probability Distributions

• A Note Not all theoretical probability
  distributions are Normal. One example of many is
  the binomial distribution.
• The binomial distribution gives the discrete
  probability distribution of obtaining exactly n
  successes out of N trials where the result of
  each trial is true with known probability of
  success and false with the inverse probability.
• The binomial distribution has a formula and
  changes shape with each probability of success
  and number of trials.
• However, in this class the normal probability
  distribution is the most useful!

a binomial distribution, used with proportions
Successes 0 1 2 3 4 5 6 7 8 9 10 11 12
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t

• We use t instead of z to be more accurate
• t curves are symmetric and bell-shaped like the
  normal distribution. However, the spread is more
  than that of the standard normal distributionthe
  tails are fatter.

Tea Tests?
df 1, 2, 3, and so on, approaching normal as df
exceeds 120.
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t

• The reason for using t is due to the fact that we
  use sample standard deviation (s) rather than
  population standard deviation (s) to calculate
  standard error. Since s, standard deviations,
  will vary from sample to sample, the variability
  in the sampling distribution ought to be greater
  than in the normal curve. t has a larger spread,
  more accurately reflecting the likelihood of
  extreme samples, especially when sample size is
  small.
• The larger the degrees of freedom (n 1 when
  estimating the mean), the closer the t curve is
  to the normal curve. This reflects the fact that
  the standard deviation s approaches s for large
  sample size n.
• Even though z-scores based on the normal curve
  will work for larger samples (n gt 120) SPSS uses
  t for all tests because it works for small
  samples and large samples alike.
• (df the number of scores that are free to vary
  when calculating a statistic . . . n - ?)

Tea Tests?
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Comparing Two Groups

• Were going to move forward to more sophisticated
  statistics, building on what we have learned
  about confidence intervals and significance
  tests.
• Sociologists look for relationships between
  concepts in the social world.
• For example
• Does ones sex affect income?
• Focus on the relationship between the concepts
  Sex and Income
• Does ones race affect educational attainment?
• Focus on the relationship between the concepts
  Race and Educational Attainment

I love sophisticated statistics!
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Comparing Two Groups

• In this section of the course, you will learn
  ways to infer from a sample whether two concepts
  are related in a population.
• Independent variable (X) That which causes
  another variable to change when it changes.
• Dependent variable (Y) That which changes in
  response to change in another variable.
• X ? Y
• (X Sex or Race) (Y Income or
  Education)
• The statistical technique you use will depend of
  the level of measurement of your independent and
  dependent variablesthe statistical test must
  match the variables!
• Levels of Measurement Nominal, Ordinal,
  Interval-Ratio

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Comparing Two Groups

• The test you choose depends on level of
  measurement
• Independent Dependent Statistical Test
• Dichotomous Interval-ratio Independent Samples
  t-test
• Dichotomous
• Nominal Nominal Cross Tabs
• Ordinal Ordinal
• Dichotomous Dichotomous
• Nominal Interval-ratio ANOVA
• Ordinal Dichotomous
• Dichotomous
• Interval-ratio Interval-ratio Correlation and
  OLS Regression
• Dichotomous

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Comparing Two Groups

• Independent Dependent Statistical Test
• Dichotomous Interval-ratio Independent Samples
  t-test
• Dichotomous
• An independent samples t-test is concerned with
  whether a mean or proportion is equal between two
  groups. For example, does sex affect income?

? Income
? Income
µ
µ
Womens mean
Mens Mean ???
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Comparing Two Groups

• Independent Samples t-tests
• Earlier, our focus was on the mean. We used the
  mean of the sample (statistic) to infer a range
  for what our population mean (parameter) might be
  (confidence interval) or whether it was like some
  guess or not (significance test).
• Now, our focus is on the difference in the mean
  for two groups. We will use the difference of
  the sample means (statistic) to infer a range for
  what our population difference in means
  (parameter) might be (confidence interval) or
  whether it is like some guess (significance test).

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Comparing Two Groups

• The difference will be calculated as such
• D-bar Y-bar2 Y-bar1
• For example
• Average Difference in Income by Sex
• Male Average Income Female Average Income
• (What would it mean if mens income minus womens
  income equaled zero?)

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Comparing Two Groups

• Like the mean, if one were to take random sample
  after random sample from two groupswith normal
  population distributionsand calculate and record
  the difference between groups each time, one
  would see the formation of a Sampling
  Distribution for D-bar that was normal and
  centered on the two populations difference.

average difference between two groups samples

Sampling
Distribution of D-bar
Z -3 -2 -1 0 1 2 3
95 Range
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Comparing Two Groups

• So the rules and techniques we learned for means
  apply to the differences in groups means.
• One creates sampling distributions to create
  confidence intervals and do significance tests in
  the same ways.
• However, the standard error of D-bar has to be
  calculated slightly differently.
• For Means
• (s1)2 (s2)2
• s.e. (s.d. of the sampling distribution)
  n1 n2
• (not assuming equal variances)
• For Proportions
• s.e. ?1 (1 - ?1) ?2
  (1 - ?2)
• n1 n2

df less than n1 n2 - 2
df n1 n2 - 2
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Comparing Two Groups

• When variances are assumed to be equal, and
  sample sizes differ, we use the pooled estimate
  of variance for the standard error.

Estimated Standard error pooled Start with a
pooled variance.                              
             Then
df n1 n2 - 2
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Comparing Two Groups

• Calculating a Confidence Interval for the
  Difference between Two Groups Means
• By slapping the sampling distribution for the
  difference over our samples difference between
  groups, D-bar, we can find the values between
  which the population difference is likely to be.
• 95 C.I. D-bar /- 1.96 (s.e.) Remember
  When
• (Y-bar2 Y-bar1) /- 1.96
  (s.e.) sample sizes are
• Or (?2 ?1) /- 1.96
  (s.e.) small, t ? z, and
• /- 1.96 may not be
• 99 C.I. D-bar /- 2.58 (s.e.) appropriate.
• (Y-bar2 Y-bar1) /- 2.58
  (s.e.)
• Or (?2 ?1) /- 2.58 (s.e.)

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Comparing Two Groups

• EXAMPLE
• We want to know what the likely difference is
  between male and female GPAs in a population of
  college students with 95 confidence.
• Sample 50 men, average gpa 2.9, s.d. 0.5
• 50 women, average gpa 3.1, s.d. 0.4
• 95 C.I. Y-bar2 Y-bar1 /- 1.96 s.e.
• Find the standard error of the sampling
  distribution
• s.e. ?(.5)2/ 50 (.4)2/50 ? .005
  .003 ? .008 0.089
• Build the width of the Interval. 95 corresponds
  with a z or t of /- 1.96.
• /- z s.e /- 1.96 0.089 /- 0.174
• Insert the mean difference to build the interval
• 95 C.I. (Y-bar2 Y-bar1) /- 1.96 s.e.
  3.1 - 2.9 /- 0.174 0.2 /- 0.174
• The interval 0.026 to 0.374

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Comparing Two Groups

• Conducting a Test of Significance for the
  Difference between Two Groups Means
• By slapping the sampling distribution for the
  difference over a guess of the difference between
  groups, Ho, we can find out whether our sample
  could have been drawn from a population where the
  difference is equal to our guess.
• Two-tailed significance test for ?-level .05
• Critical z or t /- 1.96
• To find if there is a difference in the
  population,
• Ho ?2 - ?1 0
• Ha ?2 - ?1 ? 0
• Collect Data
• Calculate z or t z or t (Y-bar2 Y-bar1)
  (?2 - µ1)
• 
  s.e.
• Make decision about the null hypothesis (reject
  or fail to reject)
• Report P-value

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Comparing Two Groups

• EXAMPLE
• We want to know whether there is a difference in
  male and female GPAs in a population of college
  students.
• Two-tailed significance test for ?-level .05
• Critical z or t /- 1.96
• To find if there is a difference in the
  population,
• Ho ?2 - ?1 0
• Ha ?2 - ?1 ? 0
• Collect Data
• Sample 50 men, average gpa 2.9, s.d. 0.5
• 50 women, average gpa 3.1, s.d. 0.4
• s.e. ?(.5)2/ 50 (.4)2/50 ? .005 .003
  ? .008 0.089
• Calculate z or t z or t 3.1 2.9
  0 0.2 2.25
• 
  0.089 0.089
• Make decision about the null hypothesis Reject
  the null. There is enough difference between
  groups in our sample to say that there is a
  difference in the population. 2.25 gt1.96
• Find P-value p or (sig.) .0122 x2 (table
  gives one-tail only) .0244
• We have a 2.4 chance that the difference in our
  sample could have come from a population where
  there is no difference between men and women.
  That chance is low enough to reject the null, for
  sure!

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Comparing Two Groups

• The steps outlined above for
• Confidence intervals
• And
• Significance tests
• for differences in means are the same you would
  use for differences in proportions.
• Just note the difference in calculation of the
  standard error for the difference.

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Comparing Two Groups

• Now lets do an example with SPSS, using the
  General Social Survey.