Exam 7: Sampling Distributions

Which of the following correctly describes the central limit theorem with respect to ?

The mean for a sample selected randomly from a population is most likely to be lower or higher than the known value of the population mean.

Suppose the average mileage rating (in miles per gallon) of a particular model car is 28 mpg with a standard deviation of 8 mpg. A random sample of 64 cars of this model is selected. Find the mean of the sampling distribution. ______________ Find the standard deviation. ______________ What is the probability that the sample mean exceeds 30 mpg? ______________

The Central Limit Theorem does not apply to the sample means of large samples drawn from a discrete distribution.

The standard deviation of a statistic that is used to estimate an unknown parameter is called the standard error of the statistic.

An accountant reviewed a firm's billing for an entire year and computed an average bill of $125, with a standard deviation of $15. The firm's comptroller claims that a sample of 50 bills would have saved a lot of work and achieved the same result. Is the sampling distribution of the sample mean approximately normal? ______________ What is the mean? ______________ What is the standard deviation? ______________ Is the comptroller's claim valid? ______________ Explain. ________________________________________________________

If the standard error of the sampling distribution of the sample proportion is 0.02049 for samples of size 500, then the population proportion must be either:

Samples of n = 100 items were selected hourly over a 100-hour period, and the sample proportion of defectives was calculated each hour. The mean of the 100 sample proportions was 0.042. Calculate the upper and lower control limits for a p chart. UCL = ______________ LCL = ______________ Explain how to construct a p chart for the process and how it can be used. ________________________________________________________

Suppose an actual census showed that 20% of the households in Michigan have incomes in excess of $60,000. Assume that a random sample of 500 households in Michigan is taken. Then, the standard error of the sampling distribution of sample proportion of households who have incomes in excess of $60,000 will be:

A group of medical researchers reported on the accuracy of using magnetic resonance imaging (MRI) to evaluate ligament sprains and tears on 50 patients. Consecutive patients with acute or chronic knee pain were selected from the clinical practice of one of the researchers and agreed to participate in the study. Describe the sampling plan used to select study participants. ________________________________________________________ What chance mechanism was used to select this sample of 50 individuals with knee pain? ______________ Can valid inferences be made using the results of this study? ______________ Why or why not? ________________________________________________________

A 1-in-k systematic random sample involves the random selection of one of the first k elements in an ordered population, and then the systematic selection of every kth element thereafter.

The spread of the distribution of sample means is considerably less than the spread of the sampled population.

The population of Pepsi cans filled by a particular machine is known to be normally distributed with a mean of 12 ounces and a standard deviation of.16 ounces. Given this information, the sampling distribution of the sample mean for a random sample of 16 cans will also be normally distributed with a mean equal to .

The expression SE ( ) represents the standard deviation of the sampling distribution of the sample proportion, .

A questionnaire was mailed to 1000 registered municipal voters selected at random. Only 600 questionnaires were returned, and of the 600 returned, 450 respondents were strongly opposed to a surcharge proposed to support the city Parks and Recreation Department. Are you willing to accept the 75% figure as a valid estimate of the percentage in the city who are opposed to the surcharge? ______________ Why or why not? ________________________________________________________

Numerical descriptive measures calculated from a sample are called Statistic.

There are four ways of selecting a distinct, unordered sample of size n = 2 without replacement from a population of size N = 4.

Identify the sampling design for each of the following. The Vice President for Academic Affairs of Southern State University decides which research proposals submitted to her by each college dean will be submitted for federal funding consideration. ______________ Every student in an introductory genetics class is assigned a number. The professor then randomly selects ten numbers and surveys the corresponding students. ______________ Class rosters list student names in alphabetical order. A professor wants to select some students to participate in a research project. The professor starts with the 5th name and then selects every 10th student thereafter. ______________ An apartment complex manager randomly selects 10 buildings from the complex of 30 buildings, and then interviews one household from every apartment in the 10 buildings. ______________ The student body of a large law school is comprised of 85% in-state residents and 15% out-of-state residents. An evaluation team wishes to interview 200 alumni from the school. They are instructed to contact 170 in-state alumni and 30 out-of-state alumni. ______________

According to the Central Limit Theorem, any sampling distribution of sample proportion normal provided np < 5, while .

In a particular large school system, the average IQ of elementary school children is 105 and the standard deviation is 15. A sample of 81 children is randomly selected from elementary schools within the system. Find the mean of the sampling distribution of . ______________ Find the standard deviation. ______________ What is the probability that the average IQ of this sample is below 108? ______________