Chapter 7 Section 2 Exercise 030 Find The Expected Count
Chapter 7 Section 2 Exercise 030find The Expected Count And The Cont
Chapter 7, Section 2, Exercise 030 requires calculating the expected count and the contribution to the chi-square statistic for a specific cell in a two-way table, namely the (Group 3, No) cell, given the observed data. The task involves understanding how to compute expected counts under the assumption of independence and how to measure each cell's contribution to the overall chi-square statistic to assess the association between variables.
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The purpose of this exercise is to reinforce understanding of chi-square tests of independence, particularly the computation of expected counts and the contribution of individual cells to the chi-square statistic. This is fundamental in determining whether there is a significant association between categorical variables, which, in this case, are grouped into different categories and yes/no responses.
The first step in solving this problem involves computing the expected counts for the specified cell (Group 3, No). The expected count is predicated on the assumption of independence between the row and column variables, meaning the probability of a particular cell's occurrence is proportional to the product of the totals for the corresponding row and column, divided by the overall total.
The formula for the expected count for a cell in a contingency table is:
Expected Count = (Row Total × Column Total) / Grand Total
Given this, precise calculations require knowing the actual observed frequencies and marginal totals for the table. As the specific observed counts and totals are not provided here, the general process involves substituting the appropriate row and column totals into the formula. For example, if the total for Group 3 is G3_total, the total for 'No' responses is No_total, and the overall total is N_total, then:
Expected Count for (Group 3, No) = (G3_total × No_total) / N_total
Once the expected count is calculated, the contribution to the chi-square statistic from this cell can be computed using the formula:
Contrib = (Observed - Expected)^2 / Expected
where 'Observed' is the actual observed frequency in that cell. This value quantifies how much that particular cell contributes to the overall chi-square statistic, indicating the degree of deviation from the expected under independence.
In practice, these calculations are conducted for all cells in the table, contributing to an overall chi-square statistic, which is then compared to the chi-square distribution to assess the independence of the variables at a chosen significance level. A larger contribution suggests a greater deviation from the expectation and potentially a significant association.
This exercise illustrates critical concepts in statistical inference regarding categorical data analysis. Understanding how to calculate and interpret expected counts and contributions enhances the ability to analyze contingency tables and draw meaningful conclusions about relationships between variables.
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