Title: Quantitative Methods
1Quantitative Methods
- Part 3
- Chi - Squared Statistic
2Recap 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
3Introduction 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
4Example 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
5Was 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
6Chi-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
7Stage1 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
8Stage 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
9Stage 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)
10Stage 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
11Stage 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
12Stage 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
13Chi-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
14Chi-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
15What 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
16Chi-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
17Chi-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
18Chi-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
19Stage 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
20Workshop
- 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