Hypothesis Testing and Comparing Two Proportions

1
Hypothesis Testing and Comparing Two Proportions

• Hypothesis Testing Deciding whether your data
  shows a real effect, or could have happened by
  chance
• Hypothesis testing is used to decide between two
  possibilities
• The Research Hypothesis
• The Null Hypothesis

2
H1 and H0

• H1 The Research Hypothesis
• The effect observed in the data (the sample)
  reflects a real effect (in the population)
• H0 The Null Hypothesis
• There is no real effect (in the population)
• The effect observed in the data (the sample) is
  just due to chance (sampling error)

3
Example Comparing Proportions

• H0 The proportions are not really different
• H1 The proportions are really different
• Example 1 Are pennies heavier on one side?
• Example 2 Do males mention footware in
  personals ads more often than females do?

4
The Logic of Hypothesis Testing

1. Assume the Null Hypothesis (H0) is true
2. Calculate the probability (p) of getting the
   results observed in your data if the Null
   Hypothesis were true
3. If that probability is low (lt .05) then reject
   the Null Hypothesis
4. If you reject the Null Hypothesis, that leaves
   only the Research Hypothesis (H1)

5

• Assume the Null Hypothesis is true
• The coins are fair (balanced)
• Calculate the probability (p) of getting the
  results observed in your data if the Null
  Hypothesis were true
• How often would you get 8/10 coins coming up
  heads if the coins were fair? You would get 8/10
  heads less than 5 of the time.
• If that probability is low (lt .05) then reject
  the Null Hypothesis
• That is unlikely, so the Null Hypothesis must be
  false.
• If you reject the Null Hypothesis, that leaves
  only the Research Hypothesis
• We conclude that the coins are not fair
  (balanced).

6
Calculating p

• How do you calculate the probability that the
  observed effect is just due to chance?
• Use a test statistic
• Are two proportions different? Chi-square
• Are two means different? t-test
• Are more than two means different? ANOVA or
  F-test

7
The Logic is Always the Same

1. Assume nothing is going on (assume H0)
2. Calculate a test statistic (Chi-square, t, F)
3. How often would you get a value this large for
   the test statistic when H0 is true? (In other
   words, calculate p)
4. If p lt .05, reject the null hypothesis and
   conclude that something is going on (H1)
5. If p gt .05, do not conclude anything.

8
Demonstrating Hypothesis Testing with Chi-square

• Example 1 Testing whether coins are unbalanced
  
• Example 2 Testing whether men are more likely
  to mention footware in personals ads than women
  are.
• (see Excel spreadsheet for both examples)

9
Assumptions of Chi-square Test

• Each observation must be INDEPENDENT one data
  point per subject
• DV is categorical (often yes/no)
• Calculations must be made from COUNTS, not
  proportions or percentages
• No cell should have an expected value of less
  than 5

10
Using Chi-square in SPSS to compare two
proportions

• Setting up the data file copy data from excel
  and paste it into SPSS data file
• Performing the Chi-square test (next slide)
• Interpreting the Results (separate slide)
• Reporting the Results (separate slide)

11
Performing the Chi-Square Test

• Name the variables using the variables tab in the
  SPSS data window
• analyze -gt descriptive statistics -gt crosstabs
• Use arrow button to move gender into rows box
• Use arrow button to move footware into
  columns box
• Click Statistics box
• Check the box for Chi-square, then click
  Continue
• Click the Cells box.
• Under Percentages check the boxes for Row and
  Column
• Click OK

12
Interpreting the Results

• Case Processing Summary look for missing
  data, etc.
• Gender x Footware Crosstabulation shows the
  counts of observations in each cell, and the
  percentages within each row and within each
  column.
• Chi-square Tests look at Pearson chi-square
  line
• Value 5.33 This is the value of Chi-square
• Asymp Sig .021 This is the p value
• Compare these values to those I calculated by
  hand on the excel spreadsheet

13
Reporting the Results

• Report the value of chi-square, the degrees of
  freedom (df), and the p value. Also mention how
  many observations there were.
• EXAMPLE A greater proportion of men than women
  mentioned footware in their ads (see Table 1).
  Of the six ads placed by men, 83 mentioned
  footware. Only 17 of the six ads placed by
  women mentioned footware. This difference was
  significant by a Chi-square test, Chi-square (1)
  5.3, p lt .05.