Hypothesis Tests, Statistical Significance

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Hypothesis Tests, Statistical Significance
Correlation Coefficients
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What is a hypothesis?
An assumption subject to verification or proof
An educated guess about what is true in the
world
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Two Types of Hypotheses

• The Null Hypothesis Ho
• There is no difference between groups.
• There is no relationship between variables.
• Ho mmales mfemales
• vs.
• The Alternative Hypothesis Ha
• There is a difference between groups.
• There is a relationship between the variables.
• Ha mmales ? mfemales (for two-tailed test)
• mmales gt or lt mfemales (for
  one-tailed test)

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Statistical Significance

• Are results due to random sampling error or
  chance? Or, is it unlikely that the results
  observed are due to chance?
• If the test results are statistically
  significant, reject the null hypothesis and
  conclude there are real differences
• The researchers decision is subject to error

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Errors in Hypothesis Testing

• Type I Errors (called alpha, or a error)
• The error of rejecting the null hypothesis when
  it is in fact true, or concluding there are
  relationships or differences when none really
  exist.
• To avoid a Type I error a conservative alpha
  level like .01 might be used.
• Type II Error (beta errors)
• The error of retaining the null hypothesis when
  it is in fact false, or
• concluding there are no relationships or
  differences when in fact they
• do exist.
• To avoid a Type II error a liberal alpha level
  such as .10 might be used

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Standard Normal Distribution
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Graphic Demonstrating Hypothesis Testing for a
Two-Tailed Test
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Steps in Statistical Significance
(Hypothesis)Testing

• STEPS
• Step 1 Set alpha level ( a ) reflects level
  of Type I error the researcher is willing to risk
• Results must have probability of error equal
  to or less than alpha before the researcher will
  reject the null hypothesis and conclude the
  results are statistically significant

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Steps in Statistical Significance Testing

• Step 2 Conduct the appropriate data analysis
  procedures for the test
• Step 3 You will get a TEST statistic
  (coefficient, chi square, t value, F value, etc.)
  that measures how close the sample has come to Ho
• Step 4 Look at the probability (p-value) of
  error associated with getting your test statistic
• Step 5 Compare the p-value to your alpha level
  

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The Research Decision

• Retain the null if the p-value is GREATER than
  your alpha level
• For example, if a .05 and p .10, retain the
  null and conclude results are NOT statistically
  significant
• Reject the null if the p-value is equal to or
  LESS than your alpha level, conclude Ha
• For example, if a .05 and p .001, reject the
  null, results are statistically significant

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Practical vs. Statistical Significance

• Results can sometimes be statistically
  significant, but the difference or strength isnt
  enough to be of any practical consequence
• Statistical significance is easier to obtain when
  sample is large (because SE is lower)

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Bivariate Measures of AssociationThe Pearson PM
Correlation Coefficient (r)

• Relationship, not causation, between 2 variables
  
• Both variables must be INTERVAL level
• Measures the direction degree of a relationship
  Direction is positive or negative
• POSITIVE The variables move in the same
  direction (i.e., when one is high (low) the other
  is high (low)
• NEGATIVE The variables move in OPPOSITE
  directions (i.e., when one is high the other is
  low)

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The Pearson PM Correlation Coefficient ( r )

• Can range from 1.00 to 1.00
• Degree Generally r gt .60 considered strong,
  between .40 and .60 moderate, r lt .40 weak (lt .10
  to 0.00 no relationship)
• A correlation of 0.00 indicates no linear
  relationship between the variables
• Coefficient of determination (r2) shows
  proportion of change in one variable explained by
  change in another (explained variance)

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Correlation Example

• Example A interpretation
• There is a strong, positive relationship between
  yrs in school and income level (r .90, p
  .000). More years in school is associated with
  higher income level.
• Example B interpretation
• There is a strong, negative relationship between
  income level and of children (r -.72, p
  .000). Lower income households tend to have more
  children in the home.
• Example C interpretation
• There is no relationship between income level and
  weight.

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1-way 2-way Chi Square (Crosstabs)
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When to use 1-Way Chi Square (c2)Goodness of
fit

• Analysis of frequency distributions to test
  whether distribution of responses fits
  presupposed proportions in population
• Univariate, the variable is categorical (nominal
  or ordinal), i.e, check only one type questions
  
• What did you like best about your dining
  experience today?

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The Statistical Question

• There will likely be differences in responses to
  the different categories.
• For example What did you like best about us
  today?
• 43 said quality of food
• 34 said fast, friendly service
• 23 said restaurant atmosphere
• But are these differences statistically
  significant? In other words, do these
  differences reflect attitudes of the population,
  or just this particular sample?

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Example 1-way Chi Square

• Research question Are Dominos, Pizza Hut, and
  Papa Johns pizza equally preferred among UNM
  students?
• Questionnaire question Which of the following
  do you prefer, Dominos, Pizza Hut, or Papa
  Johns? (check one)
• You sample 105 UNM students

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Hypothesis Testing

• Ho pd pph ppj
• (i.e., if preferences equal, expected value
  should be 35/105 or 33.3 for each)
• Ha pd ? pph ? ppj
• Assume an alpha of .05

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SPSS for 1-way Chi Square

• Analyzegtnonparametric testsgtchi square
• Move your variable to the test variable list and
  hit OK
• There is a difference in how many people actually
  said they preferred each and how many were
  expected (35)
• The test statistic, c2 2.46, df 2
• Is this a big enough difference to make an
  inference, or could differences be due to random
  sampling error?

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Conclusion

• For c2 2.46, df 2, and a .05, the pvalue
  would need to be equal to or less than .05 to be
  significant.
• p .293, Our conclusion here?
• Retain the null. There is a slight difference in
  preferences but it is not a statistically
  significant difference and could be due to random
  sampling error.

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When to use2-way Chi Square (c2)

• For compare group research questions
• Involving 2 variables, the grouping variable and
  whatever variable you wish to compare the
  subgroups on (bivariate)
• When both variables are categorical, i.e.,
  nominal or ordinal

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Example 2-way Chi Square

• Research question Are Dominos, Pizza Hut, and
  Papa Johns pizza equally preferred by male and
  female UNM students?
• Questionnaire question Which of the following
  do you prefer, Dominos Pizza Hut, or Papa
  Johns? (check one)
• AND Gender Male Female
• You sample 105 UNM students

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Significant Test for 2-way Chi Sq

• Ho Males and females have equal preferences for
  Dominos, Pizza Hut, and Papa Johns pizza.
• Ha Males and females do not have not equal
  preferences for Dominos, Pizza Hut, and Papa
  Johns pizza.
• Set alpha for testing here, a .01

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AnalyzegtDescriptive StatsgtCrosstabs

• Put one variable in Row and the other in Column
• Request observed expected counts, row, column
  percentages (dont need total) so you will have
  all the information you need to interpret each
  cell

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Interpretation

• Look at the overall test statistic (c2 14.01, p
  .001)
• Conclusion
• Do you reject the null or accept the null at a
  .01?
• Reject null. There is a significant difference
  between males females in terms of their pizza
  preferences.
• Now, interpret further by looking at the cells.
  Decide which way you are reading (down or across
  doesnt matter) and use the frequency
  percentages to describe what is different between
  the groups (NOT within a group!). Look at cells
  with big difference between observed and expected
  values.

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Interpreting Percentages for 2-way Chi square

• Always start with of the.. and insert either
  your row or column title. If you are reading
  rows, read across within the row variable. If
  you are reading columns, read down within the
  column variable
• Conclusion It appears males prefer Dominos and
  Papa Johns and females prefer Pizza Hut.
  Reading down columns Of those preferring
  Dominos, 75 are male and 25 are female. Of
  those preferring Pizza Hut, 34.1 are male and
  65.9 are female. Or read across rows Of the
  males, 45.8 prefer Dominos as opposed to only
  19.6 of females. Of the females, 58.7 prefer
  Pizza Hut as opposed to only 23.7 of males.