Chi-square test or χ 2 test student notes
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Used to test the________________________ of categorical data
•
Three types
–
__________________________________ (univariate)
–
___________________________________ (bivariate)
–
____________________________________ (univariate with two samples)
χ 2 distribution
•
Different df have different curves
•
Skewed _______________________
•
As df increases, curve shifts toward right & becomes more like a _________________________curve
Assumptions
•
SRS – reasonably random sample
•
Have _________________________ of categorical data & we expect each category to happen at least once
•
Sample size – to insure that the sample size is large enough we should expect at least_________ in each
category.
Formula
χ =∑
2
( obs − exp )
2
exp
•
Uses univariate data
•
Want to see how well the _________________________ counts “fit” what we _______________ the counts to
be
•
Use χ 2 cdf function on the calculator to find p-values
Df = number of categories -1
χ 2 Goodness of fit test
•
Uses univariate data
•
Want to see how well the _____________________counts “fit” what we ______________ the counts to be
Hypotheses – written in words
H0: proportions are ________________
Ha: at least ____________________ proportion is not the same
Be sure to write in context!
Does your zodiac sign determine how successful you will be? Fortune magazine collected the zodiac signs of 256 heads
of the largest 400 companies. Is there sufficient evidence to claim that successful people are more likely to be born
under some signs than others?
How many would you expect in each sign if there were no difference between them?
How many degrees of freedom?
χ 2 test for independence
•
Used with categorical, bivariate data from________________ sample
•
Used to see if the two categorical variables are associated (_____________________) or not associated
(___________________________)
Hypotheses – written in words
H0: two variables are independent
Ha: two variables are__________________________
Be sure to write in context!
A beef distributor wishes to determine whether there is a relationship between geographic region and cut of meat
preferred. If there is no relationship, we will say that beef preference is independent of geographic region. Suppose
that, in a random sample of 500 customers, 300 are from the North and 200 from the South. Also, 150 prefer cut A, 275
prefer cut B, and 75 prefer cut C.
If beef preference is independent of geographic region, how would we expect this table to be filled in?
North
South
Total
Cut A
150
Cut B
275
Cut C
75
Total
300
200
500
Now suppose that in the actual sample of 500 consumers the observed numbers were as follows:
North
South
Total
Cut A
100
50
150
Cut B
150
125
275
Cut C
50
25
75
Total
300
200
500
Is there sufficient evidence to suggest that geographic regions and beef preference are not independent? (Is there a
difference between the expected and observed counts?)
χ 2 test for homogeneity
•
Used with a single categorical variable from ____________ (or more) independent samples
•
Used to see if the two populations are the same (____________________________)
•
Assumptions & formula remain the same!
•
Expected counts & df are found the same way as test for independence.
•
Only change is the ____________________________!
Hypotheses – written in words
H0: the proportions for the two (or more) distributions are the ___________________________
Ha: At least one of the proportions for the distributions is _________________________________
Be sure to write in context!
The following data is on drinking behavior for independently chosen random samples of male and female students.
Does there appear to be a gender difference with respect to drinking behavior? (Note: low = 1-7 drinks/wk, moderate =
8-24 drinks/wk, high = 25 or more drinks/wk)
The following data is on drinking behavior for independently chosen random samples of male and female students.
Does there appear to be a gender difference with respect to drinking behavior? (Note: low = 1-7 drinks/wk,
moderate = 8-24 drinks/wk, high = 25 or more drinks/wk)
Men
Women
Total
None
140
186
326
Low
478
661
1139
Moderate
300
173
473
High
63
16
79
Total
981
1036
2017
Suppose, in a random sample of 75 peanut M&M’s, you get the distribution of colors as shown below. Is the
distribution of peanut M&M’s different from the distribution of plain M&M’s? The distribution for plain M&M’s is
13% red, 14% yellow, 16% green, 20% orange, 13% brown, and 24% blue.
Color
a.
Red
11
Yellow
16
Green
8
Orange
5
Brown
17
Blue
18
Identify if goodness of fit, test for independence, or Test of Homogeneity
b. null and alternative hypothesis
c.
Number of M&M’s
χ2 value and p-value
d. interpret the p-value
Here is a table showing who survived the sinking of the Titanic based on whether they were crew
members, or passengers booked in first-, second-, or third-class staterooms.
Crew
First
Second
Third
Total
Alive
212
202
118
178
710
Dead
673
123
167
528
1491
Total
885
325
285
706
2201
Is there an association between survival and type of passenger on the Titanic?
a.
Identify if goodness of fit, test for independence, or Test of Homogeneity
b. null and alternative hypothesis
c.
χ2 value and p-value
d. interpret the p-value
A researcher wants to determine if the number of minutes spent watching television per day is independent of gender.
A random sample of 315 adults was selected and the results are shown below. Is there enough evidence to conclude
that the number of minutes spent watching television per day is related to gender?
a.
Identify if goodness of fit, test for independence, or Test of Homogeneity
b. null and alternative hypothesis
c.
χ2 value and p-value
d. interpret the p-value