Understand the concept of the sample proportion 𝑝̂ as a random variable whose value varies between samples, and the formulas for the mean 𝑝 and standard deviation √𝑝(1 − 𝑝)/𝑛 of the sample proportion 𝑝̂, where 𝑛 is the sample size.

A manufacturing plant produces light globes. Quality control testing shows that 18% of globes are defective. Four technicians each take a random sample of globes and calculate the sample proportion of defective globes, \( \hat{p} \). The diagram shows the sample size \( n \) and standard deviation of \( \hat{p} \) for each technician. Which technician's sample has a standard deviation consistent with the known defect rate?

A marine biologist surveys whether fish in a particular estuary exhibit a specific migration pattern. From the population of all fish in the estuary, the true proportion that exhibit this pattern is p = 0.58. (a) Calculate the mean of the sample proportion p̂ if the biologist takes random samples of size n = 150. [1] (b) Calculate the standard deviation of the sample proportion p̂ for samples of size n = 150. Give your answer to 4 decimal places. [1] (c) Calculate the standard deviation of the sample proportion p̂ if the sample size were increased to n = 600. Give your answer to 4 decimal places. [1]

A factory quality control manager knows that historically 18% of all components produced on a particular production line are defective. Random samples of size \(n = 80\) are taken daily from this production line. (a) State the mean of the distribution of the sample proportion \(\hat{p}\) of defective components. (1 mark) (b) Calculate the standard deviation of the distribution of \(\hat{p}\), correct to four decimal places. (1 mark) (c) On one particular day, a sample of 80 components contained 10 defective items. Calculate the sample proportion for this day. (1 mark) (d) Explain whether the sample proportion calculated in part (c) is unusually low, given that it lies approximately 0.78 standard deviations below the mean. (1 mark)

A regional health authority monitors flu vaccination rates among adults aged 65 and over. The population proportion of vaccinated adults in this age group is \(p = 0.72\). The authority conducts weekly surveys by randomly selecting adults aged 65 and over. Refer to the table below showing the number of vaccinated adults in each of six weekly samples. (a) Determine the sample proportion \(\hat{p}\) for Week 3. (1 mark) (b) Calculate the mean of the sample proportion distribution, and verify that it equals the population proportion. (1 mark) (c) Calculate the standard deviation of the sample proportion distribution for the sample size used in these surveys. Express your answer correct to four decimal places. (2 marks) (d) Explain why the value of \(\hat{p}\) varies across the six weeks, even though the population proportion remains constant at \(0.72\). (1 mark)