Exam 7: Introduction to Sampling Distributions

The Fallbrook Distributing Company has a soft drink bottling plant in Plano, Texas. Based on historical records, if its filling machine is working properly, the mean fill volume per can is 12.0 ounces with a standard deviation equal to 0.13 ounce. Further, the distribution of fill amounts is known to be normally distributed. The State of Texas has a department whose job it is to check on such consumer- related processes as soft drink filling. The idea is to protect the consumer. The department arrives at the Fallbrook plant once a month on an unscheduled day. When they arrive, they randomly select n = 4 cans and carefully measure the volume in each can. If any of these cans contains less than 11.85 ounces, the plant is shut down until a full inspection of the filling process is performed. Based on this information, the probability that the plant will get shut down if it is operating properly is approximately 0.1251.

Although the concept of sampling distributions is an important concept in statistics, it is very unlikely that a decision maker will actually construct a sampling distribution in any practical business situation.

Sampling error occurs when the population parameter and the sample statistic are different.

If a sample is selected using random sampling methods, the primary reason that the sample mean might be different from the corresponding population mean is that the sample might be biased.

One of the things that the Central Limit Theorem tells us is that about half of the sample means will be greater than the population mean and about half will be less.

A sample proportion can be assumed normally distributed if np ≥ 5.

Suppose the life of a particular brand of calculator battery is approximately normally distributed with a mean of 75 hours and a standard deviation of 10 hours. If the manufacturer of the battery is able to reduce the standard deviation of battery life from 10 to 9 hours, what would be the probability that 16 batteries randomly sampled from the population will have a sample mean life of between 70 and 80 hours?

United Manufacturing and Supply makes sprinkler valves for use in residential sprinkler systems. United supplies these valves to major companies such as Rain Bird and Nelson, who in turn sell sprinkler products to retailers. United recently entered into a contract to supply 40,000 sprinkler valves. The contract called for at least 97% of the valves to be free of defects. Before shipping the valves, United managers tested 200 randomly selected valves and found 190 defect-free valves in the sample. The managers wish to know the probability of finding 190 or fewer defect-free valves if in fact the population of 40,000 valves is 97% defect-free. The probability is:

Suppose the life of a particular brand of calculator battery is approximately normally distributed with a mean of 75 hours and a standard deviation of 10 hours. What is the probability that a single battery randomly selected from the population will have a life between 70 and 80 hours?

A sampling distribution for a sample of n = 4 is normally distributed with a standard deviation equal to 5. Based on this information, the population standard deviation, σ, is equal to 10 mph.

When a population is not normally distributed, the Central Limit Theorem states that a sufficiently large sample will result in the sample mean being normally distributed.

If a population mean is equal to 200, the sample mean for a random sample selected from the population is about as likely to be higher or lower than 200.

In a particular city, the proportion of cars that would fail an air quality emissions test is thought to be 0.13. Given this, the probability that a random sample of n = 200 cars will have a sample proportion between 0.11 and 0.15 is approximately 0.60.

A sampling distribution for is the distribution of all possible sample means that could be computed from the possible samples of a given sample size.

Suppose it is known that the mean purchase price for all homes sold last year in Blacksburg, Virginia was $203,455. Recently, two studies were done on home sales prices. In the first study, a random sample of 200 homes was selected from the population. In the second study, a random sample of 60 homes was selected. Based on this information, we know that the second study would contain more sampling errors than the first study due to the smaller sample size.

A golf course in California has determined that the mean time it takes for a foursome to complete an 18 hole round of golf is 4 hours 35 minutes (275 minutes) with a standard deviation of 14 minutes. The time distribution is also thought to be approximately normal. Every month, the head pro at the course randomly selects a sample of 8 foursomes and monitors the time it takes them to play. Suppose the mean time that was observed for the sample last month was 4 hours 44 minutes (284 minutes). What is the probability of seeing a sample mean this high or higher?

The reason that a population mean and the mean of a random sample selected from that population might be different is that the sample mean is found by dividing by n-1 while the population mean is found by dividing by n.

A population has a proportion equal to 0.30. Calculate the following probabilities with n = 100. Find

An increase in sample size will tend to result in less sampling error.

The quality manager of a fortune cookie company believes that a larger than acceptable proportion of paper fortunes being used are blank. Suppose she takes a sample of 320 fortune cookies from the production line, and 15 of the paper fortunes are blank. The estimate of the proportion is approximately 0.047.