# 11.5 Comparison of the chi-square tests

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You have seen the χ 2 test statistic used in three different circumstances. The following bulleted list is a summary that will help you decide which χ 2 test is the appropriate one to use.

• Goodness-of-Fit: Use the goodness-of-fit test to decide whether a population with an unknown distribution "fits" a known distribution. In this case there will be a single qualitative survey question or a single outcome of an experiment from a single population. Goodness-of-Fit is typically used to see if the population is uniform (all outcomes occur with equal frequency), the population is normal, or the population is the same as another population with a known distribution. The null and alternative hypotheses are:
H 0 : The population fits the given distribution.
H a : The population does not fit the given distribution.
• Independence: Use the test for independence to decide whether two variables (factors) are independent or dependent. In this case there will be two qualitative survey questions or experiments and a contingency table will be constructed. The goal is to see if the two variables are unrelated (independent) or related (dependent). The null and alternative hypotheses are:
H 0 : The two variables (factors) are independent.
H a : The two variables (factors) are dependent.
• Homogeneity: Use the test for homogeneity to decide if two populations with unknown distributions have the same distribution as each other. In this case there will be a single qualitative survey question or experiment given to two different populations. The null and alternative hypotheses are:
H 0 : The two populations follow the same distribution.
H a : The two populations have different distributions.

## Chapter review

The goodness-of-fit test is typically used to determine if data fits a particular distribution. The test of independence makes use of a contingency table to determine the independence of two factors. The test for homogeneity determines whether two populations come from the same distribution, even if this distribution is unknown.

Which test do you use to decide whether an observed distribution is the same as an expected distribution?

a goodness-of-fit test

What is the null hypothesis for the type of test from [link] ?

Which test would you use to decide whether two factors have a relationship?

a test for independence

Which test would you use to decide if two populations have the same distribution?

How are tests of independence similar to tests for homogeneity?

Answers will vary. Sample answer: Tests of independence and tests for homogeneity both calculate the test statistic the same way $\sum _{\left(ij\right)}\frac{{\left(O-E\right)}^{2}}{E}$ . In addition, all values must be greater than or equal to five.

How are tests of independence different from tests for homogeneity?

## Bringing it together

1. Explain why a goodness-of-fit test and a test of independence are generally right-tailed tests.
2. If you did a left-tailed test, what would you be testing?
1. The test statistic is always positive and if the expected and observed values are not close together, the test statistic is large and the null hypothesis will be rejected.
2. Testing to see if the data fits the distribution “too well” or is too perfect.

what is the solution to this question?
hi
Dewan
hello
Learn
Dewan
The controls that are usually used are
what is math
Rushikesh
the controls that are usually used in quality controls and also controls a process is key tool used in run chat, control chat and design of experiment etc.,
Sravanthi
mean is number that occurs frequently in a giving data
That places the mode and the mean as the same thing. I'd define the mean as the ratio of the total sum of variables to the variable count, and it assigns the variables a similar value across the board.
Samsicker
what is mean
what is normal distribution
What is the uses of sample in real life
pain scales in hospital
Lisa
change of origin and scale
3. If the grades of 40000 students in a course at the Hashemite University are distributed according to N(60,400) Then the number of students with grades less than 75 =*
If a constant value is added to every observation of data, then arithmetic mean is obtained by
sum of AM+Constnt
Fazal
data can be defined as numbers in context. suppose you are given the following set of numbers 18,22,22,20,19,21
what are data
what is mode?
what is statistics
Natasha
statistics is a combination of collect data summraize data analyiz data and interprete data
Ali
what is mode
Natasha
what is statistics
It is the science of analysing numerical data in large quantities, especially for the purpose of inferring proportions in a whole from those in a representative sample.
Bernice
history of statistics
statistics was first used by?
Terseer
if a population has a prevalence of Hypertension 5%, what is the probability of 4 people having hypertension from 8 randomly selected individuals?