# Hypothesis Tecsting

**Topics:**Statistical tests, Statistics, Non-parametric statistics

**Pages:**116 (6986 words)

**Published:**March 2, 2014

Testing of Hypothesis:

(Non-parametric Tests)

Chapter-11: Testing of Hypothesis - (Non-parametric Tests)

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11.1. Chi - square ( χ )Test / Distribution

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11.1.1. Meaning of Chi - square ( χ )Test

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11.1.2. Characteristics of Chi - square ( χ )Test

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11.2. Types of Chi - square ( χ )Test / Distribution

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11.2.1. Chi - square ( χ )Test for Population Variance

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11.2.2. Chi - square ( χ )Test for Goodness-of-Fit

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11.2.3. Chi - square ( χ )Test or Independence

11.3. Analysis of Variance (ANOVA)

11.3.1. Meaning of ANOVA

11.3.2. ANOVA Approach

11.4. ANOVA Technique

11.4.1. One-way ANOVA

11.4.2. Two-way ANOVA

11.4.3. ANOVA in Latin-square Design

11.5. Other Nonparametric Techniques

Summary:

Key Terms:

Questions:

11.1. CHI-SQUARE (

) TEST /DISTRIBUTION

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11.1.1. Meaning of Chi - square ( χ )Test

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A chi-square test (also chi squared test or χ test) is any statistical hypothesis test in which the sampling distribution of the test statistic is a chi-square distribution when the null hypothesis is true, or any in which this is asymptotically true, meaning that the sampling distribution (if the null hypothesis is true) can be made to approximate a chi-square distribution as closely as desired by making the sample size large enough. The Chi-Square (

) test is the most popular non-parametric

test/methods, to test the hypothesis. The symbol

is the Greek letter “chi”. Like other hypothesis

testing procedures, the calculated value of

-test statistics is compared with its critical value to

know whether the null hypothesis is true.

Generally the chi-squared statistic summarizes the discrepancies between the expected number of times each outcome occurs (assuming that the model is true) and the observed number of times each outcome occurs, by summing the squares of the discrepancies, normalized by the expected numbers, over all the categories.

Data used in a chi-square analysis has to satisfy the following conditions 1.

2.

3.

4.

5.

Randomly drawn from the population,

Reported in raw counts of frequency,

Measured variables must be independent,

Observed frequencies cannot be too small, and

Values of independent and dependent variables must be mutually exclusive.

11.1.2. Characteristics of the Chi-Square Test:

1. It is not symmetric.

2. The shape of the chi-square distribution depends upon the degrees of freedom, just like Student’s t-distribution.

3. As the number of degrees of freedom increases, the chi-square distribution becomes more symmetric as is illustrated in Figure 1.

4. The values are non-negative. That is, the values of are greater than or equal to 0. 11.2. TYPES OF CHI-SQUARE TEST / DISTRIBUTION:

There are three types of chi-square test.

Chi square test for population variance which is used to test a hypothesis on a specific value of the population variance.

2. The Chi-square test for goodness of fit which compares the expected and observed values to determine how well an experimenter's predictions fit the data. 3. The Chi-square test for independence which compares two sets of categories to determine whether the two groups are distributed differently among the categories. 1.

11.2.1.

-test for Population Variance:

Chi square test for population variance is used to test a hypothesis on a specific value of the population variance. Statistically speaking, we test the null hypothesis H0: σ = σ0 against the research hypothesis H1: σ # σ0 where σ is the population mean and σ0 is a specific value of the population variance that we would like to test for acceptance.

If a sample of size n is taken from a population having a normal distribution, then there is a wellknown result which allows a test to be made of whether the variance of the population has a predetermined value. The test statistic t in this instance could be set to be the sum of squares about the sample mean, divided by the nominal value for the variance then t has a...

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