Hypothesis Testing Applied to Means

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Chapter 7

• Hypothesis Testing Applied to Means

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Typical Questions

• Q1 Is some sample mean different from what would
  be expected given some population distribution?
• On the face of it, this question should remind
  you of your previous fun with z-scores.
• In the case of z-scores, we asked whether some
  observation was significantly different from some
  sample mean.
• In the case of this question, we are asking
  whether some sample mean is significantly
  different from some population mean.

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Typical Questions

• Despite this apparent similarity, the questions
  are different because the sampling distribution
  of the mean (the t distribution) is different
  from the sampling distribution of observations
  the z distribution).
• In order to understand the distinction between
  the z and t-tests, we need to understand the
  Central Limit Theorem. . . .

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The Central Limit Theorem
Returning to our previous online demo
Sampling Distribution demonstration
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Testing Hypotheses about Single Means when
Population Variance is Known

• Although it is seldom the case, sometimes we know
  the variance (as well as the mean) of the
  population distribution of interest.
• In such cases, we can do a revised version of the
  z-test that takes into account the central limit
  theorem.

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Testing Hypotheses about Single Means when
Population Variance is Known

• Specifically

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Testing Hypotheses about Single Means when
Population Variance is Known

• With this formula, we can answer questions like
  the following

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Testing Hypotheses about Single Means when
Population Variance is Unknown

• Unfortunately, it is very rare that we know the
  population standard deviation.
• Instead we must use the sample standard
  deviation, s, to estimate ?.
• However, there is a hitch to this. While s2 is
  an unbiased estimator of ?2 (i.e., the mean of
  the sampling distribution of s2 equals ?2), the
  sampling distribution of s2 is positive skewed.

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Testing Hypotheses about Single Means when
Population Variance is Unknown

• Sample Variance (s2).
• This means that any individual s2 chosen from the
  sampling distribution of s2 will tend to
  underestimate ?2.
• Thus, if we used the formula that we used when ?
  was known, we would tend to get z values that
  were larger than they should be, leading to too
  many significant results.

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Testing Hypotheses about Single Means when
Population Variance is Unknown

• The solution? Use the same formula (modified to
  use s instead of ), find its distribution under
  H0, then use that distribution for doing
  hypothesis testing.
• The result

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Testing Hypotheses about Single Means when
Population Variance is Unknown

• When a t-value is calculated in this manner, it
  is evaluated using the t-table (p. 648 of the
  text) and the row for N-1 degrees of freedom.
• So, with all this in hand, we can now answer
  questions of the following type. . . .

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Testing Hypotheses about Single Means when
Population Variance is Unknown

• Example
• Lets say that the average human who
• has reached maturity is 68 tall. Im
• curious whether the average height of
• our class differs from this population
• mean. So, I measure the height of the
• 100 people who come to class one day,
• and get a mean 70 and a standard
• deviation of 5. What can I conclude?

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Testing Hypotheses about Single Means when
Population Variance is Unknown

• If we look at the t-table, we find the critical
  t-value for alpha.05 and 99 (N-1) degrees of
  freedom is 1.984.
• Since the tobt gt tcrit, we reject H0.

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Typical Questions

• Q2 Is the mean of one group significantly
  different from the mean of some other group?
• Example recall the paper airplane memory
  experiment where I asked people within three
  groups to estimate the speed of a car involved
  in an accident and, across groups, I varied the
  adjective used to described the collision
  (smashed vs. ran into vs. contacted). Did my
  manipulation affect speed estimates? That is,
  are the mean speed estimates of the various
  groups different?
• Note these tests are used in conjunction with
  continuous (i.e., measurement) data, not
  categorical data.

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Testing Hypotheses Concerning Pairs of Means
Matched Samples

• In many studies, we test the same subject on
  multiple sessions or in different test
  conditions.
• sexist profs example
• We then wish to compare the means across these
  sessions or test conditions.
• This type of situation is referred to as a pair
  wise or matched samples (or within subjects)
  design, and it must be used anytime different
  data points cannot be assumed to be independent.

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Testing Hypotheses Concerning Pairs of Means
Matched Samples

• As you are about to see, the t-test used in this
  situation is basically identical to the t-test
  discussed in the previous section, once the data
  has been transformed to provide difference scores.

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Difference Scores

• Assume we have
• some measure of
• rudeness and we
• then measure 10
• profs rudeness
• index once when
• the offending TA
• is male, and once
• when they are
• female.

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Difference Scores

• Question becomes, is the average difference score
  significantly different from 0?
• So, when we do the math

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Difference Scores

• The critical t with alpha equal .05 (two-tailed)
  and 9 (N-1) degrees of freedom is 2.262.
• Since tobt is not greater than tcrit, we can not
  reject H0.
• Thus, we have no evidence that the profs rudeness
  is difference across TAs of different genders.

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Testing Hypotheses Concerning Pairs of Means
Independent Samples

• Another common situation is one where we have two
  of more groups composed of independent
  observations.
• That is, each subject is in only one group and
  there is no reason to believe that knowing about
  one subjects performance in one of the groups
  would tell you anything about another subjects
  performance in one of the other groups.

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Testing Hypotheses Concerning Pairs of Means
Independent Samples

• In this situation we are said to have independent
  samples or, as it is sometimes called, a between
  subjects design.
• Example study to examine external biases of
  memory (i.e., when I threw all the paper
  airplanes around).

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Data from our Memory Bias Experiment

• Given Steves accident story, about how fast (in
  km/h) do you think the gray car was going when it
  ________ the side of the red car?

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Data from our Memory Bias Experiment

• There are, in fact, three different t-tests we
  can perform in this situation, comparing groups 1
  2, 13, or 23.
• For demonstration purposes, lets only worry
  about groups 1 2 for now.
• So, we could ask, do subjects in Group 1 give
  different estimates of the gray cars speed than
  subjects in Group 2?

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The Variance Sum Law

• When testing a difference between two independent
  means, we must once again think about the
  sampling distribution associated with H0.
• If we assume the means come from separate
  populations, we could simultaneously draw samples
  from each population and calculate the mean of
  each sample.

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The Variance Sum Law

• If we repeat this process a number of times, we
  could generate sampling distributions of the mean
  of each population, and a sampling distribution
  of the difference of the two means.
• If we actually did this, we would find that the
  sampling distribution of the difference would
  have a variance equal to the sum of the two
  population variances.

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The Variance Sum Law

• Now recall that when we performed a t-test in the
  situation where the population standard deviation
  was unknown, we used the formula
• Given all of the above, we can now alter this
  formula in a way that will allow us to use it in
  the independent means example.

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The Variance Sum Law

• Specifically, instead of comparing a single
  sample mean with some mean, we want to see if the
  difference between two sample means equals zero.
• Thus the numerator (top part) will change to

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The Variance Sum Law

• And, because the standard error associated with
  the difference between two means is the sum of
  each means standard error (by the variance sum
  law), the denominator of the formula changes to

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The Variance Sum Law

• Thus, the basic formula for calculating a t-test
  for independent samples is

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Pooling Variances Unequal Ns

• The previous formula is fine when sample sizes
  are equal.
• However, when sample sizes are unequal, it treats
  both of the S2 as equal in terms of their ability
  to estimate the population variance.

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Pooling Variances Unequal Ns

• Instead, it would be better to combine the s2 in
  a way that weighted them according to their
  respective sample sizes. This is done using the
  following pooled variance estimate

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Pooling Variances Unequal Ns

• Given this, the new formula for calculating an
  independent groups t-test is

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Coupla Notes

• Note 1 Using the pooled variances version of
  the t formula for independent samples is no
  difference from using the separate variances
  version when sample sizes are equal. It can have
  a big effect, however, when sample sizes are
  unequal.
• Note 2 As mentioned previously, the degrees of
  freedom associated with an independent samples t
  test is N1 N2 - 2.

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Heterogeneity of Variance

• The text book has a large section on
  heterogeneity of variance (pp 185-193) including
  lots of nasty looking formulae. All I want you
  to know is the following
• When doing a t-test across two groups, you are
  assuming that the variances of the two groups are
  approximately equal.
• If the variances look fairly different, there are
  tests that can be used to see if the difference
  is so great as to be a problem.

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Heterogeneity of Variance

• If the variances are different across the groups,
  there are ways of correcting the t-test to take
  the heterogeneity in account.
• In fact, t-tests are often quite robust to this
  problem, so you dont have to worry about it too
  much.
• Sometimes, heterogeneity is interesting.

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Hypothesis Testing with Means The Cookbook

• One Mean vs. One Population Mean
• Population variance known

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Hypothesis Testing with Means The Cookbook

• One Mean vs. One Population Mean
• Population variance unknown

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Hypothesis Testing with Means The Cookbook

• Two Means
• Matched samples
• first create a difference score, then. . . .

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Hypothesis Testing with Means The Cookbook

• Two Means
• Independent samples

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Hypothesis Testing with Means The Cookbook

• Two Means
• Independent samples. . .continued
• where
• Easy as baking a cake, right? Now for some
  examples of using these recipes to cook up some
  tasty conclusions. . . .

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Examples

• 1) The population spends an average of 8 hours
  per day working, with a standard deviation of 1
  hour. A certain researcher believes that profs
  work less hours than average and wants to test
  whether the average hours per day that profs work
  is different from the population. This
  researcher samples 10 professors and asks them
  how many hours they work per day, leading to the
  following data set
• perform the appropriate statistical test and
  state your conclusions.

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Examples

• 2) Now answer the question again except assume
  the population variance is unknown.
• 3) Does the use of examples improve memory for
  the concepts being taught? Joe Researcher tested
  this possibility by teaching 10 subjects 20
  concepts each. For each subject, examples were
  provided to help explain 10 of the new concepts,
  no examples were provided for the other 10. Joe
  then tested his subjects memory for the concepts
  and recorded how many concepts, out of 10, that
  the subject could remember. Here is the data
  (next slide)

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Examples
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Examples

• 4) Circadian rhythms suggest that young adults
  are at their physical peek in the early
  afternoon, and are at their physical low point in
  the early morning. Are cognitive factors
  affected by these rhythms? To test this question
  I bring subjects in to run a recognition
• memory experiment. Half of the
• subjects are run at 8 am, the other
• half at 2pm. I then record their
• recognition memory accuracy.
• Here are the results