Scaled sample, find minimum N

A question is this type if and only if, after an initial chi-squared test, the student is asked to find the minimum integer scaling factor N such that a larger sample with the same proportions would lead to a different conclusion at a given significance level.

3 questions · Challenging +1.1

5.06a Chi-squared: contingency tables
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CAIE FP2 2013 June Q11 OR
Challenging +1.8
A researcher is investigating the relationship between the political allegiance of university students and their childhood environment. He chooses a random sample of 100 students and finds that 60 have political allegiance to the Alliance party. He also classifies their childhood environment as rural or urban, and finds that 45 had a rural childhood. The researcher carries out a test, at the \(10 \%\) significance level, on this data and finds that political allegiance is independent of childhood environment. Given that \(A\) is the number of students in the sample who both support the Alliance party and have a rural childhood, find the greatest and least possible values of \(A\). A second random sample of size \(100 N\), where \(N\) is an integer, is taken from the university student population. It is found that the proportions supporting the Alliance party from urban and rural childhoods are the same as in the first sample. Given that the value of \(A\) in the first sample was 29, find the greatest possible value of \(N\) that would lead to the same conclusion (that political allegiance is independent of childhood environment) from a test, at the \(10 \%\) significance level, on this second set of data.
CAIE FP2 2012 June Q10
11 marks Standard +0.8
Random samples of employees are taken from two companies, \(A\) and \(B\). Each employee is asked which of three types of coffee (Cappuccino, Latte, Ground) they prefer. The results are shown in the following table.
CappuccinoLatteGround
Company \(A\)605232
Company \(B\)354031
Test, at the 5% significance level, whether coffee preferences of employees are independent of their company. [7] Larger random samples, consisting of \(N\) times as many employees from each company, are taken. In each company, the proportions of employees preferring the three types of coffee remain unchanged. Find the least possible value of \(N\) that would lead to the conclusion, at the 1% significance level, that coffee preferences of employees are not independent of their company. [4]
CAIE FP2 2012 June Q10
11 marks Standard +0.8
Random samples of employees are taken from two companies, \(A\) and \(B\). Each employee is asked which of three types of coffee (Cappuccino, Latte, Ground) they prefer. The results are shown in the following table.
CappuccinoLatteGround
Company \(A\)605232
Company \(B\)354031
Test, at the 5\% significance level, whether coffee preferences of employees are independent of their company. [7] Larger random samples, consisting of \(N\) times as many employees from each company, are taken. In each company, the proportions of employees preferring the three types of coffee remain unchanged. Find the least possible value of \(N\) that would lead to the conclusion, at the 1\% significance level, that coffee preferences of employees are not independent of their company. [4]