Exam 10: Sampling Distributions

The central limit theorem is basic to the concept of statistical inference, because it permits us to draw conclusions about the population based strictly on sample data, and without having any knowledge about the distribution of the underlying population.

The expected value of the sampling distribution of the sample mean $\bar { X }$ equals the population mean $\mu$ :

If a random variable X is not known to be normally distributed, has a mean of size 8 and a variance of size 1.5, describe the sampling distribution of the sample mean for samples of size 30.

If all possible samples of size n are drawn from an infinite population with a mean of 60 and a standard deviation of 8, then the standard error of the sample mean equals 1.0 only for samples of size 64.

If the daily demand for boxes of mineral water at a supermarket is normally distributed with a mean of 47.6 boxes and a standard deviation of 5.8 boxes, what is the probability that the average demand for a sample of 10 supermarkets will be less than 50 boxes in a given day?

An auditor knows from past history that the average accounts receivable for a company is $521.72, with a standard deviation of $584.64. If the auditor takes a simple random sample of 100 accounts, what is the probability that the mean of the sample will be within $120 of the population mean?

Which of the following best describes the Central Limit Theorem?

If all possible samples of size n are drawn from a population, the probability distribution of the sample mean $\bar { X }$ is called the:

Which of the following best describes the sampling distribution of the sample proportion?

The following table gives the number of pets owned for a population of four families. Family Number of pets owned 2 1 4 3 Find the sampling distribution of $\bar { X }$ .

Given an infinite population with a mean of 75 and a standard deviation of 12, the probability that the mean of a sample of 36 observations, taken at random from this population, exceeds 78 is:

A video rental store wants to know what proportion of its customers are under 21 years old. A simple random sample of 500 customers is taken, and 350 of them are under 21. Assume that the true population proportion of customers aged under 21 is 0.68. Describe the sampling distribution of proportion of customers who are under age 21.

Suppose that the average annual income of a lawyer is $150 000 with a standard deviation of $40 000. Assume that the income distribution is normal. a. What is the probability that the average annual income of a sample of 5 lawyers is more than $120 000? b. What is the probability that the average annual income of a sample of 15 lawyers is more than $120 000?

It is know that 40% of voters in a certain electorate are in favour of a particular candidate. If a sample of size 30 is taken, what is the probability that less than 35% are in favour of this political candidate?

The amount of time spent by Australian adults playing sports per day is normally distributed, with a mean of 4 hours and standard deviation of 1.25 hours. Find the probability that if four Australian adults are randomly selected, their average number of hours spent playing sport is more than 5 hours per day.

A video rental store wants to know what proportion of its customers are under 21 years old. A simple random sample of 500 customers is taken, and 350 of them are under 21. Assume that the true population proportion of customers aged under 21 is 0.68. If another simple random sample of size 500 is taken, what is the probability that more than 70% of customers are under 21 years old?

The following table gives the number of pets owned for a population of four families. Family Number of pets owned 2 1 4 3 Samples of size 2 will be drawn at random from the population. Use your answers to the previous question to calculate the mean and the standard deviation of the sampling distribution of the sample means.

The heights of 9-year-old children are normally distributed, with a mean of 123 cm and a standard deviation of 10 cm. a. Find the probability that one randomly selected 9-year-old child is taller than 125 cm. b. Find the probability that three randomly selected 9-year-old children are taller than 125 cm. c. Find the probability that the mean height of three randomly selected 9-year-old children is greater than 125 cm.

In a given year, the average annual salary of an Rugby player was $205 000, with a standard deviation of $24 500. If a simple random sample of 50 players is taken, what is the probability that the sample mean will be less than $210 000?

A manufacturing company is concerned about the number of defective items produced by their assembly line. In the past they have had 5% of their products produced defectively. They take a random sample of 35 products. What is the probability that more than 5 products in the sample are defective?