David, a first year college student, is currently

Pablo Dennis

Pablo Dennis

Answered question

2022-03-26

David, a first year college student, is currently completing his statistics assignment but he is not understanding the notes about chi-square distribution.
c. Explain the following steps involved in a chi-square test of independence:
i) Identify the claim then state the null and alternative hypotheses (HoandHa) for this test.
ii) Select the level of significance.
iii) Calculate the test statistic.
iv) Formulate the Decision Rule and Make a Decision.
v) Interpret your decision in terms of the claim.

Answer & Explanation

Charlie Haley

Charlie Haley

Beginner2022-03-27Added 14 answers

Chi-Square test of independence :
The main properties of a Chi-Square test of independence are:
1. The distribution of the test statistic is the Chi-Square distribution, with (r1)×(c1)(r1)×(c1) degrees of freedom, where r is the number of rows and c is the number of columns
2. The Chi-Square distribution is one of the most important distributions in statistics, together with the normal distribution and the F-distribution
3. The Chi-Square test of independence is right-tailed.
Following are the steps to perform the test of independence :
State the Hypothesis :
Suppose that Variable A has r levels, and Variable B has c levels. The null hypothesis states that knowing the level of Variable A does not help you predict the level of Variable B. That is, the variables are independent.
H0: Variable A and Variable B are independent.
Hα: Variable A and Variable B are not independent.
Analysis :
The analysis plan describes how to use sample data to accept or reject the null hypothesis. The plan should specify the following elements.
1. Significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.
2. Test method. Use the chi-square test for independence to determine whether there is a significant relationship between two categorical variables.
1. Degrees of freedom. The degrees of freedom (DF) is equal to:
DF=(r1)(c1)
where r is the number of levels for one categorical variable, and c is the number of levels for the other categorical variable.
2. Expected frequencies. The expected frequency counts are computed separately for each level of one categorical variable at each level of the other categorical variable. Compute rc expected frequencies, according to the following formula.
Er,c=nr×ncn
where Er,c is the expected frequency count for level r of Variable A and level c of Variable B, nr is the total number of sample observations at level r of Variable A, nc is the total number of sample observations at level c of Variable B, and n is the total sample size.
3. Test statistic. The test statistic is a chi-square random variable (χ2) defined by the following equation.
χ2=(Or,cEr,c)Er,c
where Or,c is the observed frequency count at level r of Variable A and level c of Variable B, and Er,c is the expected frequency count at level r of Variable A and level c of Variable B.
4. P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a chi-square, use the chi square table to assess the probability associated with the test statistic. Use the degrees of freedom computed above.
5. Interpret the results : If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level, and rejecting the null hypothesis when the P-value is less than the significance level.

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