Understand that there are variations in confidence intervals between samples and that most, but not all, confidence intervals contain 𝑝.

A researcher conducts multiple random samples of size $n = 200$ from a large population to estimate the population proportion $p$ of adults who support a proposed policy. For each sample, a 95% confidence interval is calculated. Which statement correctly describes the expected outcome?

A market research team conducts a survey to estimate the proportion of shoppers who prefer online purchasing. They survey a random sample of 250 shoppers and find that 135 prefer online purchasing. Using a $z$-score of 1.96, calculate the approximate 95% confidence interval for the population proportion. Give your answer to 3 decimal places.

A researcher surveys three independent random samples of voters from the same population regarding support for a proposed initiative. For each sample of 200 voters, a 95% confidence interval (using $z = 1.96$) for the population proportion $p$ is calculated. The sample proportions obtained are: $\hat{p}_1 = 0.48$, $\hat{p}_2 = 0.52$, and $\hat{p}_3 = 0.50$. (a) Calculate the margin of error $E$ for a single sample. [1] (b) Determine the confidence interval for sample 2. [1] (c) If the true population proportion is $p = 0.51$, state which of the three confidence intervals contain this value. [1]

A researcher conducts a survey to estimate the proportion of households in a region that own at least one electric vehicle. She takes 100 random samples, each of size 200 households, and calculates a 95% confidence interval for the population proportion from each sample. Which of the following statements best describes what should occur?