Use repeated random sampling data, for a variety of values of 𝑝 and a range of sample sizes, to examine the distribution of 𝑝̂ and the approximate standard normality of 𝑝̂−𝑝 √𝑝̂(1−𝑝̂)/𝑛, where the closeness of the approximation depends on both 𝑛 and 𝑝.

A political polling organisation claims that 75% of voters support a particular policy. Assuming this claim is true, find the probability that, in a random sample of 100 voters, less than 68% support the policy.

A manufacturing company inspects quality by sampling items from a production batch. The true proportion of defective items in the batch is $p = 0.15$. Four different quality-control teams independently conduct repeated random sampling, each team taking samples of the same size from the batch and calculating the sample proportion $\hat{p}$ of defective items. Which team's distribution of $\hat{p}$ would be expected to most closely approximate a normal distribution?

A pharmaceutical company claims that 72% of patients using their new medication experience symptom relief within 48 hours. A research team conducts four independent sampling experiments with different sample sizes, as shown in the table below. a) For Experiment 2, calculate the value of the standardised test statistic z = (p̂ - p) / √(p(1-p)/n), where p = 0.72 is the claimed population proportion. (2 marks) b) For Experiment 4, determine the standard deviation of the sampling distribution of p̂. (1 mark) c) Compare Experiments 1 and 3. Explain which experiment's sample proportion would be expected to provide better evidence about the closeness of the normal approximation to the distribution of p̂, and justify your answer with reference to both sample size and the value of p. (2 marks)

A researcher conducts repeated random sampling of size n from a population with proportion p of defective items. Which combination of n and p will produce a distribution of the sample proportion that most closely approximates a normal distribution?

A population has proportion \( p = 0.8 \). Two students each take random samples and calculate the sample proportion \( \hat{p} \). Student A uses a sample of size \( n = 25 \), while Student B uses a sample of size \( n = 400 \). State which student's sample proportion is expected to be closer to the true population proportion. Explain your answer in one sentence.