Quantitative Methods

1
Quantitative Methods

• Part 3
• Chi - Squared Statistic

2
Recap on T-Statistic

• It used the mean and standard error of a
  population sample
• The data is on an interval or scale
• Mean and standard error are the parameters
• This approach is known as parametric
• Another approach is non-parametric testing

3
Introduction to Chi-Squared

• It does not use the mean and standard error of a
  population sample
• Each respondent can only choose one category
  (unlike scale in T-Statistic)
• The expected frequency must be greater than 5 for
  the test to succeed.
• If any of the categories have less than 5 for the
  expected frequency, then you need to increase
  your sample size

4
Example using Chi-Squared

• Is there a preference amongst the UW student
  population for a particular web browser? (Dr C
  Prices Data)
• They could only indicate one choice
• These are the observed frequencies responses from
  the sample

Firefox IExplorer Safari Chrome Opera
Observed frequencies 30 6 4 8 2
5
Was it just chance?

• How confident am I?
• Was the sample representative of all UW students?
• Was it just chance?
• Chi-Squared test for significance
• Some variations on test
• Simplest is Null Hypothesis
• The students show no preference for a
  particular browser

6
Chi-Squared Goodness of fit (No
preference)

• The students show no preference for a
  particular browser
• This leads to Hypothetical or Expected
  distribution of frequency
• We would expect an equal number of respondents
  per category
• We had 50 respondents and 5 categories

Firefox IExplorer Safari Chrome Opera
Expected frequencies 10 10 10 10 10
Expected frequency table
7
Stage1 Formulation of Hypothesis

• There is no preference in the underlying
  population for the factor suggested.
• There is a preference in the underlying
  population for the factors suggested.
• The basis of the chi-squared test is to compare
  the observed frequencies against the expected
  frequencies

8
Stage 2 Expected Distribution

• As our null- hypothesis is no preference, we
  need to work out the expected frequency
• You would expect each category to have the same
  amount of respondents
• Show this in Expected frequency table
• Has to have more than 5 to be valid

Firefox IExplorer Safari Chrome Opera
Expected frequencies 10 10 10 10 10
9
Stage 3a Level of confidence

• Choose the level of confidence (often 0.05)
• 0.05 means that there is 5 chance that
  conclusion is chance
• 95 chance that our conclusions are certain

Stage 3b Degree of freedom

• We need to find the degree of freedom
• This is calculated with the number of categories
• We had 5 categories, df 5-1 (4)

10
Stage 3 Critical value of Chi-Squared

• In order to compare our calculated chi-square
  value with the critical value in the
  chi-squared table we need
• Level of confidence (0.05)
• Degree of freedom (4)
• Our critical value from the table 9.49

11
Stage 4 Calculate statistics

• We compare the observed against the expected for
  each category
• We square each one
• We add all of them up

Firefox IExplorer Safari Chrome Opera
Observed 30 6 4 8 2
Expected 10 10 10 10 10
52
12
Stage 5 Decision

• Can we reject the That students show no
  preference for a particular browser?
• Our value of 52 is way beyond 9.49. We are 95
  confident the value did not occur by chance
• So yes we can safely reject the null hypothesis
• Which browser do they prefer?
• Firefox as it is way above expected frequency of
  10

13
Chi-Squared No Difference from a Comparison
Population.

• RQ Are drivers of high performance cars more
  likely to be involved in accidents?
• Sample n 50 and Market Research data of
  proportion of people driving these categories
• Once null hypothesis of expected frequency has
  been done, the analysis is the same as no
  preference calculation

High Performance Compact Midsize Full size
FO 20 14 9 9
MR 10 40 30 20
FE 5 (10 of 50) 20 15 10
14
Chi-Squared test for Independence.

• What makes computer games fun?
• Review found the following
• Factors (Mastery, Challenge and Fantasy)
• Different opinion depending on gender
• Research sample of 50 males and 50 females

Mastery Challenge Fantasy
Male 10 32 8
Female 24 8 18
Observed frequency table
15
What is the research question?

• A single sample with individuals measured on 2
  variables
• RQ Is there a relationship between fun factor
  and gender?
• HO There is no such relationship
• Two separate samples representing 2 populations
  (male and female)
• RQ Do male and female players have different
  preferences for fun factors?
• HO Male and female players do not have
  different preferences

16
Chi-Squared analysis for Independence.

• Establish the null hypothesis (previous slide)
• Determine the critical value of chi-squared
  dependent on the confidence limit (0.05) and the
  degrees of freedom.
• df (R 1)(C 1) 1 2 2 (R2, C3)
• Look up in chi-squared table
• Chi-squared value 5.99

Mastery Challenge Fantasy
Male 10 32 8
Female 24 8 18
17
Chi-Squared analysis for Independence.

• Calculate the expected frequencies
• Add each column and divide by types (in this case
  2)
• Easier if you have equal number for each gender
  (if not come and see me)

Mastery Challenge Fantasy Respondents
Male (FO) 10 32 8 50
Female (FO) 24 8 18 50
Cat total 34 40 26

Male (FE) 17 20 13
Female (FE) 17 20 13
18
Chi-Squared analysis for Independence.

• Calculate the statistics using the chi-squared
  formula
• Ensure you include both male and female data

Mastery Challenge Fantasy
Male (FO) 10 32 8
Female (FO) 24 8 18

Male (FE) 17 20 13
Female (FE) 17 20 13
19
Stage 5 Decision

• Can we reject the null hypothesis?
• Our value of 24.01 is way beyond 5.99. We are 95
  confident the value did not occur by chance
• Conclusion We are 95 confident that there is a
  relationship between gender and fun factor
• But else can we get from this?
• Significant fun factor for males Challenge
• Significant fun factor for females Mastery and
  Fantasy

Mastery Challenge Fantasy
Male (FO) 10 32 8
Female (FO) 24 8 18

Male (FE) 17 20 13
Female (FE) 17 20 13
20
Workshop

• Work on Workshop 7 activities
• Your journal (Homework)
• Your Literature Review (Complete/update)

References

• Dr C. Prices notes 2010
• Gravetter, F. and Wallnau, L. (2003) Statistics
  for the Behavioral Sciences, New York West
  Publishing Company