Understand and use the relationship between margin of error, level of confidence and sample size.

A health researcher surveys a random sample of adults to estimate the proportion who complete at least 150 minutes of moderate physical activity per week. In a pilot sample of 280 adults, 182 reported meeting this guideline. The researcher plans a follow-up study and wants to be 99% confident that the sample proportion estimate will be within ±0.03 of the true population proportion. (a) Determine the sample proportion from the pilot study. (1 mark) (b) Establish an equation in \( n \) that relates the margin of error to the required sample size. (1 mark) (c) Calculate the minimum sample size \( n \) required. (1 mark) (d) State a reasonable value for the sample size, giving a brief reason. (1 mark)

A technology company wishes to estimate the proportion of its app users who have enabled push notifications. A pilot study of \(180\) randomly selected users found that \(153\) had enabled push notifications. (a) Determine the sample proportion from the pilot study. [1 mark] (b) The company plans a larger survey and requires \(98\%\) confidence that the estimate will be within \(\pm 0.025\) of the true population proportion. Establish an equation in \(n\) for the required sample size. [1 mark] (c) Solve the equation to determine the value of \(n\). [1 mark] (d) State the minimum sample size required for the survey. [1 mark]

The margin of error for estimating a population proportion is given by \(E = z\sqrt{\frac{p(1-p)}{n}}\) where \(z\) is the critical value, \(p\) is the sample proportion, and \(n\) is the sample size. If the sample size is reduced by half while keeping all other factors constant, what happens to the margin of error?

A market research company estimates that the proportion of households in a region that use online grocery services is \[\hat{p} = 0.48\] They want to conduct a new survey with a margin of error of \[E = \pm 0.03\] at a 95% confidence level. (a) Using the formula \(E = z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), establish an equation in \(n\). [1 mark] (b) Determine the required sample size \(n\), rounding to the nearest whole number. [1 mark] (c) If the company wishes to reduce the margin of error to \(\pm 0.02\), explain how this affects the required sample size and why. [1 mark]