Chapter 13 Testing Hypotheses

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Chapter 13 Testing Hypotheses

• Overview
• Research and null hypotheses
• One and two-tailed tests
• Errors
• Testing the difference between two means
• t tests

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Overview
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Overview
You already know how to deal with two nominal
variables
Independent Variables
Nominal Interval
Considers how a change in a variable affects a
discrete outcome
Lambda
Dependent Variable
Interval Nominal
Considers the difference between the mean of one
group on a variable with another group
Considers the degree to which a change in one
variable results in a change in another
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Overview
You already know how to deal with two nominal
variables
Independent Variables
Nominal Interval
Considers how a change in a variable affects a
discrete outcome
Lambda
Dependent Variable
Interval Nominal
Considers the degree to which a change in one
variable results in a change in another
Confidence Intervals t-test
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General Examples
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Specific Examples
Do people living in rural communities live longer
than those in urban or suburban areas? Do
students from private high schools perform better
in college than those from public high schools?
Is the average number of years with an employer
lower or higher for large firms (over 100
employees) compared to those with fewer than 100
employees?
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Testing Hypotheses

• Statistical hypothesis testing A procedure that
  allows us to evaluate hypotheses about population
  parameters based on sample statistics.
• Research hypothesis (H1) A statement reflecting
  the substantive hypothesis. It is always
  expressed in terms of population parameters, but
  its specific form varies from test to test.
• Null hypothesis (H0) A statement of no
  difference, which contradicts the research
  hypothesis and is always expressed in terms of
  population parameters.

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Research and Null Hypotheses

• One Tail specifies the hypothesized direction
• Research Hypothesis
• H1 ?2 ???1, or ?2 ???1 gt 0
• Null Hypothesis
• H0 ?2 ???1, or ?2 ???1 0
• Two Tail direction is not specified (more
  common)
• Research Hypothesis
• H1 ?2 ?1, or ?2 ???1 0
• Null Hypothesis
• H0 ?2 ???1, or ?2 ???1 0

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One-Tailed Tests

• One-tailed hypothesis test A hypothesis test in
  which the alternative is stated in such a way
  that the probability of making a Type I error is
  entirely in one tail of a sampling distribution.
• Right-tailed test A one-tailed test in which
  the sample outcome is hypothesized to be at the
  right tail of the sampling distribution.
• Left-tailed test A one-tailed test in which the
  sample outcome is hypothesized to be at the left
  tail of the sampling distribution.

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Two-Tailed Tests

• Two-tailed hypothesis test A hypothesis test in
  which the region of rejection falls equally
  within both tails of the sampling distribution.

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Probability Values

• Z statistic (obtained) The test statistic
  computed by converting a sample statistic (such
  as the mean) to a Z score. The formula for
  obtaining Z varies from test to test.
• P value The probability associated with the
  obtained value of Z.

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Probability Values
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Probability Values

• Alpha ( ) The level of probability at which
  the null hypothesis is rejected. It is customary
  to set alpha at the .05, .01, or .001 level.

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Five Steps to Hypothesis Testing

• Making assumptions
• Stating the research and null hypotheses and
  selecting alpha
• Selecting the sampling distribution and
  specifying the test statistic
• Computing the test statistic
• Making a decision and interpreting the results

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Type I and Type II Errors

• Type I error (false rejection error)?the
  probability (equal to ?) associated with
  rejecting a true null hypothesis.
• Type II error (false acceptance error)?the
  probability associated with failing to reject a
  false null hypothesis.

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t Test

• t statistic (obtained) The test statistic
  computed to test the null hypothesis about a
  population mean when the population standard
  deviation is unknown and is estimated using the
  sample standard deviation.
• t distribution A family of curves, each
  determined by its degrees of freedom (df). It is
  used when the population standard deviation is
  unknown and the standard error is estimated from
  the sample standard deviation.
• Degrees of freedom (df) The number of scores
  that are free to vary in calculating a statistic.

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t distribution
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t distribution table
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t-test for difference between two means
Is the value of ?2 ???1 significantly different
from 0? This test gives you the answer If
the t value is greater than 1.96, the difference
between the means is significantly different from
zero at an alpha of .05 (or a 95 confidence
level).
?The difference between the two means ? the
estimated standard error of the difference
The critical value of t will be higher than 1.96
if the total N is less than 122. See Appendix C
for exact critical values when N lt 122.
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Estimated Standard Error of the difference
between two meansassuming unequal variances
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t-test and Confidence Intervals
The t-test is essentially creating a confidence
interval around the difference score. Rearranging
the above formula, we can calculate the
confidence interval around the difference between
two means
If this confidence interval overlaps with zero,
then we cannot be certain that there is a
difference between the means for the two samples.
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Why a t score and not a Z score?

• Use of the Z distribution has assumes the
  population standard error of the difference is
  known. In practice, we have to estimate it and so
  we use a t score.
• When N gets larger than 50, the t distribution
  converges with a Z distribution so the results
  would be identical regardless of whether you used
  a t or Z.
• In most sociological studies, you will not need
  to worry about the distinction between Z and t.

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t-Test Example 1
Mean pay according to gender N Mean
Pay S.D. Women 46 10.29 .8766 Men 54 10.06 .90
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What can we conclude about the difference in
wages?
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t-Test Example 2
Mean pay according to gender N Mean
Pay S.D. Women 57 9.68 1.0550 Men 51 10.32 .94
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What can we conclude about the difference in
wages?
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In-Class Exercise
Using the income data from the 1991 GSS,
calculate a t test statistic to determine if the
difference between the two group means is
statistically significant.