Sampling and Estimation

State, with a reason, whether you agree with Nikki's friend.

Nikki chooses an appropriate random sample of 60 students. She finds that 45 of these students think that the sports facilities are good.

Calculate an approximate 95% confidence interval for the proportion of students who think that the sports facilities are good.

For a different investigation, Nikki uses another large random sample to calculate a 99% confidence interval and an $x$% confidence interval.

The width of the 99% confidence interval is double the width of the $x$% confidence interval.

Calculate the value of $x$.

Find $\text{P}(\bar{X} > 6.2)$. You are not expected to use a continuity correction.

State why the Central Limit Theorem was needed in the calculation in part (a).

Calculate unbiased estimates of the population mean, $\mu$, and population variance, $\sigma^2$.

Calculate a 95% confidence interval for $\mu$.

Another random sample of $n$ cups of coffee is taken, where $100 < n < 120$. A 95% confidence interval for $\mu$ is calculated using this sample. You may assume that, for large samples, unbiased estimates of $\sigma^2$ are very similar.

Without calculation, state whether this confidence interval would be wider or narrower than the confidence interval found in part (a)(ii). Give a reason for your answer.

By considering probabilities, show that the choice made by this method is not random.

Later, Emma has to choose two people at random from three people.

Describe how Emma could use a single throw of a fair six-sided dice to make this random choice.

Find a 95% confidence interval for the population mean of $X$.

State which part of your solution to part (a) makes use of the Central Limit Theorem.

Maroulla’s first random number is 0.401.

Find the member number that is produced by this random number.

A member number of 680.

A member number of 850.

Explain briefly how your answers to part (b) show that Maroulla’s method does not produce a random sample.