# 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.

A consumer advocate agency wants to estimate the mean repair cost of a washing machine. the agency randomly selects 40 repair cost and find the mean to be $100.00.The standards deviation is$17.50. Construct a 90% confidence interval for the mean.
pls I need understand this statistics very will is giving me problem
Sixty-four third year high school students were given a standardized reading comprehension test. The mean and standard deviation obtained were 52.27 and 8.24, respectively. Is the mean significantly different from the population mean of 50? Use the 5% level of significance.
No
Ariel
how do I find the modal class
look for the highest occuring number in the class
Kusi
the probability of an event occuring is defined as?
The probability of an even occurring is expected event÷ event being cancelled or event occurring / event not occurring
Gokuna
what is simple bar chat
Simple Bar Chart is a Diagram which shows the data values in form of horizontal bars. It shows categories along y-axis and values along x-axis. The x-axis displays above the bars and y-axis displays on left of the bars with the bars extending to the right side according to their values.
statistics is percentage only
the first word is chance for that we use percentages
it is not at all that statistics is a percentage only
Shambhavi
I need more examples
how to calculate sample needed
mole of sample/mole ratio or Va Vb
Gokuna
how to I solve for arithmetic mean
Yeah. for you to say.
James
yes
niharu
how do I solve for arithmetic mean
niharu
add all the data and divide by the number of data sets. For example, if test scores were 70, 60, 70, 80 the total is 280 and the total data sets referred to as N is 4. Therfore the mean or arthritmatic average is 70. I hope this helps.
Jim
*Tan A - Tan B = sin(A-B)/CosA CosB ... *2sinQ/Cos 3Q = tan 3Q - tan Q
standard error of sample
what is subjective probability
how to calculate the Steadman rank correlation
David
what is sampling? i want to know about the definition of sampling.