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Mathematical Methods — Question 20

QCAA 2020, Paper 2 · 5 marks

Q20 · 2020 · Technology-activeComplex unfamiliar5 marks

QUESTION 20 (5 marks)

Assuming the approximate normality of sample proportions (p^1(\hat p_1 and p^2)\hat p_2) and based on two independent samples, the approximate confidence interval for the difference of two proportions is given by (p^1−p^2−zp^1(1−p^1)n1+p^2(1−p^2)n2,  p^1−p^2+zp^1(1−p^1)n1+p^2(1−p^2)n2).\left(\hat p_1-\hat p_2-z\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2}},\;\hat p_1-\hat p_2+z\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2}}\right). If the approximate confidence interval for the difference between two proportions does not contain 0, this provides evidence that the two proportions are not equal. The data in the table shows the observed frequencies of two drink preferences for independent samples of people who live in Town A and Town B.
TownTeaCoffeeTotal
A111105216
B150107257
Using the approximate 99% confidence interval for the difference of two proportions, determine if there is evidence to conclude that drink preference is associated with the town where the person lives.
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