Consider the population $\{1,4,6,8\}$ representing the number of days patients stayed in a rehabilitation center. Suppose random samples of size $n=2$ are selected with replacement. After listing all possible samples, calculate the mean of each sample and construct a table showing the sampling distribution of the sample mean. Combine sample means that are the same.

Why must both np and nq be at least 5 to use the normal approximation for the sampling distribution of p̂?

If X is the number of successes in n trials and p̂ = X/n, what is the relationship between their distributions?

A poll shows 65% of voters support a policy. In a sample of 120 voters, what is the probability that between 70 and 85 voters (inclusive) support the policy? Use the normal approximation with continuity correction.

Which of the following scenarios is most appropriate for using the normal approximation to the binomial distribution?

A quality control manager inspects 80 items and finds 12 defective. If the historical defect rate is 10%, is this sample result consistent with the historical rate? (Use the sampling distribution of p̂.)

A manufacturer claims that 95% of its products are defect-free. In a random sample of 200 products, what is the probability that at least 190 are defect-free? Use the normal approximation with continuity correction.

A fair coin is flipped 60 times. What is the probability that the sample proportion of heads is between 0.4 and 0.6?

Suppose $55\%$ of adults in a city have a gym membership. In a random sample of $120$, is it unusual to find $50$ or fewer with a gym membership? Use the $z$-scores.

Consider the population $\{2,6,14\}$. Calculate the mean of the sampling distribution of the median if samples of size $n=2$ are randomly selected with replacement.