Exam 10: Sampling Distributions

In order to estimate the mean salary for a population of 500 employees, the managing director of a certain company selected at random a sample of 40 employees. a. Would you use the finite population correction factor in calculating the standard error of the sample mean? Explain. b. If the population standard deviation is $800, compute the standard error both with and without using the finite population correction factor. c. What is the probability that the sample mean salary of the employees will be within ±$200 of the population mean salary?

An infinite population has a mean of 33 and a standard deviation of 6. A sample of 100 observations is to be taken at random from this population. The probability that the sample mean will be between 34.5 and 36.1 is:

A sample of size 35 is taken from a normal population, with mean of 65 and standard deviation of 9.3. Describe the sampling distribution of the sample mean.

The heights of women in Australia are normally distributed, with a mean of 165 centimetres and a standard deviation of 10 centimetres. What is the probability that the mean height of a random sample of 30 women is smaller than 162 centimetres?

The standard deviation of a sampled population is also called the standard error of the sample mean.

Random samples of size 64 are taken from an infinite population whose mean is 160 and standard deviation is 32. The mean and standard error of the sample mean, respectively, are:

A researcher conducted a survey on a university campus for a sample of 64 third-year students and reported that third-year students read an average of 3.12 books in the prior academic semester, with a standard deviation of 2.15 books. Determine the probability that the sample mean is: a. less than 3.45. b. between 3.38 and 3.58. c. above 2.94.

The sampling distribution of the sample proportion has mean p and variance pq/n.

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, all four play sport for more than 5 hours per day.

The sampling distribution of the sample proportion is approximately normal provided that np ≥ 5 and nq ≥ 5

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 375 of them are under 21. Assume that the true population proportion of customers aged under 21 is 0.68. What is the probability that the sample proportion will be within 0.03 of the true proportion of customers who are aged under 21?

If a random sample of size n is drawn from a normal population, then the sampling distribution of the sample mean $\bar { X }$ will be:

When a great many simple random samples of size n are drawn from a population that is normally distributed, the sampling distribution of the sample means will be normal, regardless of sample size n.

A population that consists of 250 items has a mean of 37 and a standard deviation of 13. A sample of size 5 is taken at random from this population. The standard error of the sample mean equals (up to three decimal places):

An infinite population has a mean of 100 and a standard deviation of 20. Suppose that the population is not normally distributed. What does the central limit theorem say about the sampling distribution of the mean if samples of size 64 are drawn at random from this population?

A normally distributed population with 200 elements has a mean of 60 and a standard deviation of 10. The probability that the mean of a sample of 25 elements taken from this population will be smaller than 56 is:

A sample of size n is selected at random from an infinite population. As n increases the standard error of the sample mean decreases.

A sample of 30 observations is drawn from a normal population with mean of 750 and a standard deviation of 300. Suppose the population size is 600. a. Find the expected value of the sample mean $\bar { X }$ . b. Find the standard error of the sample mean $\bar { X }$ . c. Find P( $\bar { X }$ > 790). d. Find P( $\bar { X }$ < 650). e. Find P(760 < $\bar { X }$ < 810).

The heights of women in Australia are normally distributed, with a mean of 165 centimetres and a variance of 100 centimetres². A random sample of five women is selected. What is the probability that the sample mean is greater than 162 centimetres?

Consider an infinite population with a mean of 160 and a standard deviation of 25. A random sample of size 64 is taken from this population. The standard deviation of the sample mean equals: