Sampling Distributions (College Board AP® Statistics): Exam Questions

Let be the population proportion and be the size of a random sample taken from the population. Which of the following cases has the greatest standard deviation for the sampling distribution of the sample proportion, ?

A sociologist wants to estimate the average number of hours workers spend commuting to work each day in a particular city. They plan to take a random sample of 500 workers and ask them the question: How many hours do you spend commuting to work each day? Let represent the population mean (the mean number of hours spent commuting each day for all workers in the city) and let represent the sample mean (the mean number of hours spent commuting each day for the 500 randomly selected workers). Is the sample mean, , an unbiased estimator of the population mean, ?

In a particular country, 65% of people under the age of 20 would vote for a new president, while 60% of those aged 20 or over would do the same. A simulation was performed in which a random sample of 300 people under the age of 20 and 270 people aged 20 or over were asked if they would vote for a new president. The difference in the sample proportions of those who answered yes, , was found. The simulation was run 5,000 times, and the sampling distribution of the difference in sample proportions is shown below. Which of the following , , and values are the most likely?

It is known that 12% of new cell phones from a certain company break after six months. A quality control inspector follows a random sample of 250 customers who bought new phones from the company for six months to see if they have broken or not. What is the best approximation to the probability that less than 10% of the 250 phones sampled are broken? Note that z represents the standard normal variable.