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CHI SQUARE

The CHI SQUARE Statistic

Tests for Goodness of Fit and Independence

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CHI SQUARE

Preview Color is known to affect human moods

and emotion. Sitting in a pale-blue room is more calming than sitting in a bright-red room

Based on the known influence of color, Hill and Barton (2005) hypothesized that the color of uniform may influence the outcome of physical sports contest

The study does not produce a numerical score for each participant. Each participant is simply classified into two categories (winning or losing)

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CHI SQUARE

Preview The data consist of frequencies or

proportions describing how many individuals are in each category

This study want to use a hypothesis test to evaluate data. The null hypothesis would state that color has no effect on the outcome of the contest

Statistical technique have been developed specifically to analyze and interpret data consisting of frequencies or proportions CHI SQUARE

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CHI SQUARE PARAMETRIC AND NONPARAMETRIC STATISTICAL

TESTS The tests that concern parameter and

require assumptions about parameter are called parametric tests

Another general characteristic of parametric tests is that they require a numerical score for each individual in the sample. In terms of measurement scales, parametric tests require data from an interval or a ratio scale

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CHI SQUARE PARAMETRIC AND NONPARAMETRIC STATISTICAL

TESTS Often, researcher are confronted with

experimental situation that do not conform to the requirements of parametric tests. In this situations, it may not be appropriate to use a parametric test because may lead to an erroneous interpretation of the data

Fortunately, there are several hypothesis testing techniques that provide alternatives to parametric test that called nonparametric tests

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CHI SQUARE

NONPARAMETRIC TEST Nonparametric tests sometimes are

called distribution free tests One of the most obvious differences

between parametric and nonparametric tests is the type of data they use

All the parametric tests required numerical scores. For nonparametric, the subjects are usually just classified into categories

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CHI SQUARE

NONPARAMETRIC TEST Notice that these classification involve

measurement on nominal or ordinal scales, and they do not produce numerical values that can be used to calculate mean and variance

Nonparametric tests generally are not as sensitive as parametric test; nonparametric tests are more likely to fail in detecting a real difference between two treatments

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CHI SQUARE THE CHI SQUARE TEST FOR GOODNESS

OF FIT… uses sample data to test hypotheses

about the shape or proportions of a population distribution. The test

determines how well the obtained sample proportions fit the population

proportions specified by the null hypothesis

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CHI SQUARE THE NULL HYPOTHESIS FOR THE GOODNESS

OF FIT For the chi-square test of goodness of fit,

the null hypothesis specifies the proportion (or percentage) of the population in each category

Generally H0 will fall into one of the following categories:○No preference

H0 states that the population is divided equally among the categories

○No difference from a Known populationH0 states that the proportion for one population are not different from the proportion that are known to exist for another population

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CHI SQUARE THE DATA FOR THE GOODNESS OF FIT

TEST Select a sample of n individuals and count how

many are in each category The resulting values are called observed

frequency (fo) A sample of n = 40 participants was given a

personality questionnaire and classified into one of three personality categories: A, B, or C

Category A

Category B

Category C

15 19 6

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CHI SQUARE

EXPECTED FREQUENCIES The general goal of the chi-square test for

goodness of fit is to compare the data (the observed frequencies) with the null hypothesis

The problem is to determine how well the data fit the distribution specified in H0 – hence name goodness of fit

Suppose, for example, the null hypothesis states that the population is distributed into three categories with the following proportion

Category A Category B Category C

25% 50% 25%

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CHI SQUARE

EXPECTED FREQUENCIES To find the exact frequency expected

for each category, multiply the same size (n) by the proportion (or percentage) from the null hypothesis

25% of 40 = 10 individual in category A50% of 40 = 20 individual in category B25% of 40 = 10 individual in category C

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CHI SQUARE

THE CHI-SQUARE STATISTIC The general purpose of any

hypothesis test is to determine whether the sample data support or refute a hypothesis about population

In the chi-square test for goodness of fit, the sample expressed as a set of observe frequencies (fo values) and the null hypothesis is used to generate a set of expected frequencies (fe values)

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CHI SQUARE

THE CHI-SQUARE STATISTIC The chi-square statistic simply

measures ho well the data (fo) fit the hypothesis (fe)

The symbol for the chi-square statistic is χ2

The formula for the chi-square statistic is

χ2 = ∑ (fo – fe)2

fe

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CHI SQUARE

A researcher has developed three different design for a computer keyboard. A sample of n = 60 participants is obtained, and each individual tests all three keyboard and identifies his or her favorite.The frequency distribution of preference is: Design A = 23, Design B = 12, Design C = 25.Use a chi-square test for goodness of fit with α = .05 to determine whether there are significant preferences among three design

LEARNING CHECK

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CHI SQUARE THE CHI-SQUARE TEST FOR

INDEPENDENCE The chi-square may also be used to test whether there is a relationship between two variables

For example, a group of students could be classified in term of personality (introvert, extrovert) and in terms of color preferences (red, white, green, or blue).

RED WHITE GREEN

BLUE ∑

INTRO

10 3 15 22 50

EXTRO

90 17 25 18 150

100 20 40 40 200

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CHI SQUARE OBSERVED AND EXPECTED

FREQUENCIESfo RED WHITE GREEN BLUE ∑

INTRO 10 3 15 22 50

EXTRO

90 17 25 18 150

∑ 100 20 40 40 200RED WHITE GREEN BLUE ∑

INTRO

EXTRO∑

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CHI SQUARE

THE CHI-SQUARE STATISTIC

χ2 = ∑ (fo – fe)2

fe