Understand and use the approximate confidence interval, (𝑝̂ − 𝑧√𝑝̂(1−𝑝̂) 𝑛, 𝑝̂ + 𝑧√𝑝̂(1−𝑝̂) 𝑛), as an interval estimate for 𝑝, the population proportion, where 𝑧 is the appropriate quantile for the standard normal distribution.

A local library conducted a survey to determine whether to extend evening opening hours. From a random sample of 80 library members, 28 members supported the extended hours proposal. (a) Determine the sample proportion of members who support the proposal. Express your answer as a simplified fraction. [1 mark] (b) Using \(z = 1.96\) for a 95% confidence level, calculate the approximate margin of error. Give your answer correct to three decimal places. [2 marks] (c) Determine the approximate 95% confidence interval for the proportion of all library members who support extended evening hours. Express the endpoints as decimals correct to three decimal places. [1 mark]

A medical research team is investigating the effectiveness of a new treatment protocol across two independent hospital networks. In Network X, \(142\) out of \(280\) patients showed improvement after the treatment. In Network Y, \(165\) out of \(300\) patients showed improvement. Using the approximate \(95\%\) confidence interval for the difference of two proportions, determine if there is evidence to conclude that treatment effectiveness differs between the two hospital networks. For a \(95\%\) confidence interval, use \(z = 1.96\).

A health survey randomly sampled 250 workers and found that 85 workers reported regularly practising mindfulness exercises. Which of the following represents the correct pair of values ($\hat{p}$ and $n$) needed to calculate an approximate 95% confidence interval for the proportion of all workers who practise mindfulness?

A manufacturing plant inspects a random sample of 400 components and finds that 72 are defective. An approximate 95% confidence interval for the true proportion of defective components is calculated using \( z = 1.96 \). Which interval is correct?