Exam 8: Sampling Distributions

When a great many simple random samples are drawn from a population that is normally distributed,the same means will also be normally distributed only when the sample size is greater than 30.

For each of the following combinations of n and $\pi$ ,recall the rule of thumb used to indicate when the normal distribution is a good approximation of the sampling distribution for the sample proportion p,and determine whether the rule is satisfied. A)n = 50; $\pi$ = 0.05 B)n = 50; $\pi$ = 0.10 C)n = 100; $\pi$ = 0.05 D)n = 100; $\pi$ = 0.99

Find the mean and standard deviation of the sampling distribution of the mean. Mean = ____________________ SD = ____________________

The standard error is:

The standard deviation of the sampling distribution of the mean is referred to as the ____________________.

A sample of 50 observations is drawn at random from a normal population whose mean and standard deviation are 75 and 6,respectively. What does the central limit theorem say about the sampling distribution of the sample mean? Why? Find the mean and standard error of the sampling distribution of the sample mean. Mean = ____________________ SD = ____________________ C) $\text { Find } P ( \bar { x } > 73 )$ D) $\text { Find } P ( \bar { x } < 74 )$

Briefly,what is the difference between a standard deviation and a standard error?

The sample proportion can be considered a random variable that will take on different values as the procedure leading to the sample proportion is repeated a great many times.

In a selected year,the top 50 performers in men's professional tennis averaged 24.04 years of age with a standard deviation of 3.63 years.A random sample of ten is taken.Compute the standard deviation of $\bar { x }$ .

Vehicles Four families live on a cul-de-sac represent a very small population.The number of vehicles owned by each family is shown below: Family = Number of vehicles Ali 1 Brown 1 Cosby 2 Davis 3 -List all possible simple random samples of size two that can be taken from this population and determine the mean number of vehicles for each sample.

The chairman of a statistics department in a certain college believes that 70% of the department's graduate assistantships are given to international students.A random sample of 50 graduate assistants is taken.Assume that the chairman is correct and that $\pi$ = 0.70.What is the probability that the sample proportion p will be within $\pm$ 0.05 of the population proportion $\pi$ ?

A sample of size 400 is drawn from a population whose mean and variance are 5,000 and 10,000,respectively.Find the following probabilities: A) $P ( \bar { x } < 4,990 )$ ____________________ B) $P ( 4,995 < \bar { x } < 5,010 )$ ____________________ C) $P ( \bar { x } = 5,000 )$ ____________________

A manufacturer of printers claims that their color cartridge will print an average of 200 4-by-6 photos with a standard deviation of 60 photos.A random sample of 40 customers counted the number of photos printed with one cartridge.If the manufacturer's claim is correct,what is the probability that the average customer printed at least 210 pages with one cartridge?

A manufacturer of galvanized chains advertises the load capacity of a 16-foot length of chain has an average of 2650 lbs.and a standard deviation of 520 lbs.If a retailer receives an order of 75 chains,what is the probability that the sample mean will be greater than 2775 lbs?

The chairman of a statistics department in a certain college believes that 70% of the department's graduate assistantships are given to international students.A random sample of 50 graduate assistants is taken.Assume that the chairman is correct and that $\pi$ = 0.70.What is the sampling distribution of the sample proportion p? Explain.

In order to estimate the mean salary for a population of 500 employees,the president of a certain company selected at random a sample of 40 employees.What is the probability that the sample mean salary of the employees will be within $\pm$ $200 of the population mean salary?

A manufacturing company packages peanuts for Piedmont Airlines.The individual packages weigh 1.4 grams with a standard deviation of 0.6 grams.For a flight of 152 passengers receiving the peanuts,the probability that the average weight of the packages is less than 1.3 grams is:

Briefly,what is the difference between a probability distribution and a sampling distribution?

An auditor knows from past history that the average accounts receivable for ABC Company is $441.27 with a standard deviation of $555.48.If the auditor takes a sample of 100 accounts,what is the probability that the mean of the sample will be within $100 of the population mean?

List all possible samples of size n = 2 from the five individuals and determine the mean for each sample.