Exam 8: Sampling Methods and the Central Limit Theorem

Manufacturers were subdivided into groups by volume of sales. Those with more than $100 million in sales were classified as large; those from $50 to $100 million as medium size; and those between $25 and $50 million, and so on. Samples were then selected from each of these groups. What is this type of sampling called?

Bones Brothers & Associates prepare individual tax returns. Over prior years, Bones Brothers has maintained careful records regarding the time to prepare a return. The mean time to prepare a return is 90 minutes and the standard deviation of this distribution is 14 minutes. Suppose 100 returns from this year are selected and analyzed regarding the preparation time. What is the probability that the mean time for the sample of 100 returns for this year is greater than 92?

If a population is normally distributed, then the sampling distribution of the sample means will be __________________________.

A population consists of the following four values: 8, 10, 12, and 16. From this population, there are six different samples of size 2. The means of the six samples of size 2 are 9, 10, 12, 11, 13, and 14. Compute the mean of the distribution of the sample means and the population mean. What is true about the two values?

For a given population, the mean of all the sample means of sample size n, and the mean of all (N) population observations (X) are _______.

The Office of Student Services at a large western state university maintains information on the study habits of its full-time students. Their studies indicate that the mean amount of time undergraduate students study per week is 20 hours. The hours studied follows the normal distribution with a standard deviation of six hours. Suppose we select a random sample of 144 current students. What is the probability that the mean of this sample is between 19.25 hours and 21.0 hours?

Suppose a research firm conducted a survey to determine the mean amount steady smokers spend on cigarettes during a week. A sample of 100 steady smokers revealed that the sample mean is $20 and the sample standard deviation is $5. What is the probability that a sample of 100 steady smokers spend between $19 and $21?

LongLast Inc. produces car batteries. The mean life of these batteries is 60 months. The distribution of the battery life closely follows the normal probability distribution with a standard deviation of eight months. As a part of its testing program, LongLast tests a sample of 25 batteries. What proportion of the samples will have a mean useful life less than 56 months?

A university has 1,000 computers available for students to use. Each computer has a 250-gigabyte hard drive. The university wants to estimate the space occupied on the hard drives. A random sample of 100 computers showed a mean of 115 gigabytes used with a standard deviation of 20 gigabytes. What is the probability that a sample mean is greater than 120 gigabytes?

Mileage tests were conducted on a randomly selected sample of 100 newly developed automobile tires. The mean tread wear was 50,000 miles, with a standard deviation of 3,500 miles. What is the best estimate of the average tread life in miles for the entire population of these tires?

A population consists of the following four values: 8, 10, 12, and 16. From this population, there are six different samples of size 2. The means of the six samples of size 2 are 9, 10, 12, 11, 13, and 14. Compare the dispersion in the population with that of the sample means.

A university has 1,000 computers available for students to use. Each computer has a 250-gigabyte hard drive. The university wants to estimate the space occupied on the hard drives. A random sample of 100 computers showed a mean of 115 gigabytes used with a standard deviation of 20 gigabytes. What is the probability that a sample mean is greater than 200 gigabytes?

Based on the central limit theorem, the sample mean can be used as a good estimator of the population mean, assuming that the size of the sample is sufficiently large.

The mean of all the sample means is _______.

When computing probabilities for the sampling distribution of the sample mean, the z-statistic is computed as .

The tread life of tires mounted on light duty trucks follows the normal probability distribution with a mean of 60,000 miles and a standard deviation of 4,000 miles. Suppose you bought a set of four tires, what is the likelihood the mean tire life of these four tires is more than 66,000 miles?

The American Auto Association reports that the mean price per gallon of regular gasoline is $3.10, with a population standard deviation of $0.20. Assume a random sample of 16 gasoline stations is selected and their mean cost for regular gasoline is computed. What is the probability that the sample mean is between $2.98 and $3.12?

The Intelligence Quotient (IQ) test scores for adults are normally distributed with a mean of 100 and a standard deviation of 15. What is the probability we could select a sample of 50 adults and find the mean of this sample is between 95 and 105?

The items or individuals of the population are arranged in a file drawer alphabetically by date received. A random starting point is selected and then every k^th member of the population is selected for the sample. This sampling method is called systematic random sampling.

The difference between the population mean and a sample mean is called ________.