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STAM4000

Quantitative Methods

Week 11

Chi-square tests

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UP until now:

• Hypothesis tests of the population mean, μ, (TESTS FOR ONE QUANTITATIVE

VARIABLE) based on NORMAL DISTRIBUTIONS of either Z or t.

• Hypothesis tests in linear regression (TESTS FOR TWO OR MORE QUANTITATIVE

VARIABLE) – this was actually using the NORMAL distribution of t.

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Today:

• Hypothesis tests about TWO CATEGORICAL VARIABLES – specifically, we will be

testing for “INDEPENDENCE” – this uses the Chi-square distribution, that is positively

skewed or skewed to the right.

• Way back in Week 3, “probability” the independence between only TWO EVENTS.

• In Week 11, we will concentrate on INDEPENDENCE OF TWO CATEGORICAL

VARIABLES OVERALL.

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5 ts #1 #2 #3 Introduction to Chi-square tests Chi-square test of independence Standardized (Z score) Chi-square residuals (difference/deviation) |
Week 11 Chi-square tes Learning Outcomes |

AssignmentTutorOnline

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Why does this matter?

If we have

categorical

variables, and our

data are counts (or

frequencies), we

can still examine

whether variables

are independent. https://www.reddit.com/r/mathmemes/comments/b2dub1/poor_souls/

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#1 Introduction to Chi-square tests

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Note:

As with all calculated test statistics, we can be given a p-value.

However, it can be difficult to find the p-value for a Chi-square calculated value using

the Chi-square statistical tables.

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#1 What are we testing here?

Chi-square tests are about one or more categorical variables.

We will follow the familiar process of hypothesis testing:

• Check conditions, but now we will have conditions for Chi-square.

• Follow the steps of hypothesis testing:

o Write hypotheses: now we will have “names of our categorical variables

included”

o Find the Chi-square calculated test statistic, Chi-square value from a formula

o Find the Chi-square critical value, from Chi-square statistical tables

o Sketch a Chi-square curve, positively skewed or skewed to the right

o Decision, Comparison Chi-square calc test statistic WITH Chi-square critical

value

o Conclusion, ties our decision to the original question

Chi-square

is read as

“ki square”

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Ho, is our “Null hypothesis” which we ASSUME TO BE TRUE.

Ha, is our “Alternative hypothesis”, which we try to gather evidence and PROVE is

NOW true.

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#1 Three different type of Chi-square tests

•Compares the observed distribution of one categorical

variable, to an expected distribution of that categorical

variable.

Goodness-of-fit test

•Compares the distribution of several groups for the same

Test of homogeneity categorical variable

•Examines the difference between observed and expected

counts of two categorical variables, to determine if there

is an association between the two variables.

Test of independence

We will cover the test of independence and standardized residuals in

STAM4000

Chi-square

is read as

“ki square”

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#1

We assume:

•The outcome of each of the identical trials would fall into one of two categories.

•The probability of these outcomes is constant throughout the experiment.

•If p is the probability of success, the Expected frequency of an event X with

success rate p is E[X] = np

o The expected frequencies are calculated, assuming the null hypothesis, Ho,

is TRUE.

Chi-square tests: Theory

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#1

•Our test compares the observed frequencies, from the sample, with the

expected frequencies, from the hypothesised model in Ho.

•We ask:

“Is the difference between what we expected and what we observed,

due to sampling variability or is the differences large enough to be

due to a change from the hypothesis model in Ho?”

•We square the difference between the observed and expected frequencies, to

make them positive AND then we divide this by the expected frequency, to get

an idea of the relative size of the difference.

Theory continued …

Chi-square calculator link:

http://www.socscistatistics.com/tests/chisquare2/Default2.aspx

Chi-square notes link: archive.bio.ed.ac.uk/jdeacon/statistics/tress9.html

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#1 Chi-square calculated test statistic

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