COMPARISON OF MEANS

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COMPARISON OF MEANS

• AED 616
• SPRING 2007

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Introduction

• We will consider a frequently encountered
  problem how to compare the means of two samples
  for statistical significance.
• Lets consider two examples.

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Example 1

• A researcher wants to determine whether there are
  differences between men and women voters in their
  attitudes toward welfare. Samples of men and
  women are drawn at random and administered an
  attitude scale to obtain a score for each
  subject. Means for the two samples were computed.
  Women had a mean of 40.00 (on a scale of 0 to
  50, where 50 is the most favorable). Men had mean
  of 35.00 The researcher wants to determine
  whether there is a significant difference between
  men and women. What accounts for the 5-point
  difference? One possible explanation is the null
  hypothesis, which states that there is no true
  differences between men and women that the
  observed difference is due to sampling errors
  created by random sampling.

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• Example 1 illustrates that two means may be
  obtained from a survey.

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Example 2

• A random sample of kittens is fed a vitamin
  supplement from birth to see if the supplement
  increases their visual acuity. Another random
  sample is fed a placebo that looks like the
  supplement but contains no vitamins. A the end of
  the study, both samples are tested for visual
  acuity and an average acuity score is calculated
  for each sample. The kittens that took the
  supplement score 4 points higher on the average
  than did the control group. What accounts for the
  4-point difference? One possible explanation is
  the null hypothesis, which states that there is
  no true difference between the two samples of
  kittens tht the observed difference is due to
  sampling errors created by random sampling.

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• Example 2 illustrates that two means may be
  obtained from an experiment a study in which
  treatments are given in order to observe for
  their effects.
• Surveys and experiments are very frequently
  conducted, and they often yield two means each,
  so you can see how important it is to be able to
  test the null hypothesis for the difference
  between the two sample means.
• A statistician named William Gosset developed the
  t-test for exactly the situations we are
  considering.
• As a test of the null hypothesis, it yields a
  probability that a given null hypothesis is
  correct.
• When the probability that it is correct is low
  say .05 or 5 or less we usually reject the
  null hypothesis.
• What leads the t test to give us a low
  probability that the null hypothesis is correct?
  Here are three basic factors

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Factor 1

• The larger the samples, the less likely the
  difference between two means was created by
  sampling errors. Larger samples have less
  sampling error than smaller ones. Thus, when
  large samples are used, the t test is more likely
  to yield a probability low enough to allow us to
  reject the null hypothesis than when small
  samples are used.

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Factor 2

• The larger the differences between the two means,
  the less likely the difference was created by
  sampling errors. Random sampling tends to create
  many small differences and few large ones. Thus,
  when large differences between means are
  obtained, the t test is more likely to yield a
  probability low enough to allow us to reject the
  null hypothesis than when small differences are
  obtained.

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Factor 3

• The smaller the variance among the subjects, the
  less likely that the difference between the two
  means was created by sampling errors. To
  understand this, consider a population in which
  everyone is identical they al look alike, think
  alike, and speak and act in unison. How many do
  you have to sample to get a good sample? Only
  one, because the are all the same. Thus, when
  there is no variation among subjects, it is not
  possible to have sampling errors. As the
  variation increases, sampling errors are more and
  more likely to occur.

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Types of t tests

• There are two types of t tests
• One is for Independent data (sometimes called
  uncorrelated data)
• Other is for dependent data (sometimes correlated
  data).
• Examples 1 and 2 have independent data.
• Example 3 (to follow) describes a study with
  dependent data.

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Example 3

• In a study of visual acuity, same-sex siblings
  (two brothers or two sisters) were identified for
  a study. For each pair of siblings, a coin was
  tossed to determine which one received a vitamin
  supplement and which one received a placebo.
  Thus, in the control group, there is a subject
  who is a same-sex sibling of each subject in the
  experimental group.

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• The means that results from the study in Example
  3 are subject to less error than the means from
  Example 2.
• Remember that in Example 2, there was no matching
  or pairing of subjects before assignment to
  conditions.
• In Example 3, the matching of subjects assure us
  that the two groups are more similar than if just
  tow independent samples were used.
• To the extent that genetics and gender are
  associated with visual acuity, the two groups in
  Example 3 will be more similar at the onset of
  the experiment than the two groups in Example 2.
• The t test for dependent data takes the possible
  reduction for error into account. Thus, it is
  important to select the right t test.

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• Independent data are obtained when there is no
  matching or pairing of subjects across groups.
• Here is how to compute t for independent data and
  how to interpret it using the t table.
• Formula for t is simple
• m1 m2
• t --------------
• SDm
• Where
• m1 is the mean of the group with the
  higher mean
• m2 is the mean of the group with the
  lower mean
• SDm is the standard error of the
  difference between means

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• Numerator is the difference between the two
  means. Larger the difference, the larger the
  value of t.
• Denominator starts with the symbol S (for
  standard deviation). The subscripts (D for
  difference and m for means) tell us that it is
  the standard deviation of the difference between
  means this standard deviation is called the
  standard error of the difference between means.
• We want to know whether the difference between
  two means is an unlikely event. If it is unlikely
  (for example, likely to occur less than 5 times
  in 100 due to chance alone), then we will declare
  the difference to be statistically significant
  that is, unlikely to be the result of random
  errors.

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• It is impractical to directly obtain the SDm for
  a given t test. What we do is estimate it given
  what we know about the sample size and the
  variance of the samples (the variance is simply
  the square of the standard deviation, whose
  symbol is s2)using the following formula (check
  out the board)

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• We call the result the observed value of t.
• In this example, we observed a value of 3.300. To
  evaluate its meaning, we need to take account of
  the number of cases that underlie it, using this
  formula for the degrees of freedom (df)3
• df n1 n2 2
• Where
• n1 is the number of cases in Group1
• n2 is the number of cases in Group 2
• 2 is a constant for this type of problem

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• For our example
• df 12 11 2 23 2 21
• To evaluate, we use the appropriate critical
  value of t found in the t table in backs of stats
  books.
• Examining Table 4, we look up the degrees of
  freedom (21 for our example).
• Look up 21 in the first column, then look to
  the right of the .05 column. There you find the
  critical value of 2.080.
• We have found that for 21 degrees of freedom,
  only values as extreme as 2.080 are unlikely
  events at the .05 level.
• Our observed value is 3.300. Is this an unlikely
  event?
• Yes. Because the observed value of 3.300 exceeds
  the critical value for the .01 level for 21
  degrees of freedom, which is 2.831.
• Thus we can reject the null hypothesis and
  declare the result to be statistically
  significant at the .01 level.

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Reporting the Results of t Tests

• We are considering the use of the t test to
  measure the difference between two sample means
  for significance.
• Obviously, you should report the values of the
  means before reporting the results of the test on
  them.
• In addition, you should report the values of the
  standard deviations and number of cases in each
  group.
• This may be done within the context of a sentence
  or in a table.

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• The way it may be reported in the research
  literature is
• The difference between the means is statistically
  significant (t 3.22, df 10, p
  test)