QCE Vault / Mathematical Methods Mathematical Methods — Question 8 QCAA 2022, Paper 1 · 1 mark
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QUESTION 8 In a survey, 80 respondents exercised daily, while 120 did not. When calculating the approximate 95% confidence interval for the proportion of people who exercise daily, the margin of error is
(A) 1.96 0.4 ( 1 − 0.4 ) 200 1.96\sqrt{\dfrac{0.4(1-0.4)}{200}} 1.96 200 0.4 ( 1 − 0.4 ) (B) 0.95 0.4 ( 1 − 0.4 ) 200 0.95\sqrt{\dfrac{0.4(1-0.4)}{200}} 0.95 200 0.4 ( 1 − 0.4 ) (C) 1.96 0.67 ( 1 − 0.67 ) 120 1.96\sqrt{\dfrac{0.67(1-0.67)}{120}} 1.96 120 0.67 ( 1 − 0.67 ) (D) 0.95 0.67 ( 1 − 0.67 ) 120 0.95\sqrt{\dfrac{0.67(1-0.67)}{120}} 0.95 120 0.67 ( 1 − 0.67 ) WORKED SOLUTION
Answer A 1 mark ANSWER A — 1.96 0.4 ( 1 − 0.4 ) 200 1.96\sqrt{\dfrac{0.4(1-0.4)}{200}} 1.96 200 0.4 ( 1 − 0.4 ) Worked solution
p ^ = 80 / 200 = 0.4 \hat p=80/200=0.4 p ^ = 80/200 = 0.4 and the approximate 95% margin of error is 1.96 p ^ ( 1 − p ^ ) / n 1.96\sqrt{\hat p(1-\hat p)/n} 1.96 p ^ ( 1 − p ^ ) / n , giving A. Worked explanation by QCE Vault; correct option verified against the QCAA answer key.
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