Q. What do you mean by
expected frequencies in (a) chi-square test for testing independence of
attributes, and (b) chi-square test for testing goodness-of-fit? Also explain
the procedure you follow in calculating the expected values in each of the
above situations.
The chi-square
test is a fundamental statistical test used to examine whether observed data
fits a certain distribution or if two categorical variables are independent. In
both the chi-square test for independence and the chi-square test for
goodness-of-fit, expected frequencies are a crucial element. Understanding how
expected frequencies are calculated and how they play into the overall
hypothesis testing process is vital. Below is a detailed explanation of what
expected frequencies mean in both contexts, followed by the steps involved in
their calculation.
Chi-Square
Test for Testing Independence of Attributes:
In the chi-square
test for independence, the goal is to determine whether there is a significant
association between two categorical variables, typically arranged in a
contingency table. The test is based on comparing the observed frequencies in
each category with the frequencies we would expect to see if the variables were
independent.
Expected
Frequencies in Chi-Square Test for Independence:
The expected
frequency in this context represents the frequency that would occur in each
cell of the contingency table if the two variables were independent. These
expected frequencies are based on the assumption that the distribution of one
variable does not depend on the distribution of the other. In other words, the
expected frequencies reflect what we would expect to observe if the null
hypothesis (that the two variables are independent) were true.
Formula
for Expected Frequencies:
For a contingency
table with r rows and c columns, the expected
frequency for a cell in the i-th row and j-th column is calculated using
the formula:
Where:
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