Exam 7: Sampling Distributions

If a process is in control, we expect all the data values to fall within three standard deviations of the mean.

If random samples of size n = 36 are drawn from a non-normal population with finite mean = 75 and standard deviation = 15, then the sampling distribution of the sample mean is approximately normally distributed with mean = 75 and standard deviation = 2.5.

The Central Limit Theorem states that the sampling distribution of the population mean, , is approximately normal, provided that .

In most processes, the process control limits are set to correspond with the specification limits on the process.

The total area under a probability density function curve is equal to 1.0.

Weights of Chocolate Bars Narrative A candy bar factory is pouring chocolate into moulds to cool. The finished bars are sold as 35 gram bars. The company will lose money if the moulds are overfilled. If the moulds are underfilled, the weight of the candy bars will be less than the wrapper label says, and the company will be subject to a fine for misrepresenting the size of its product. The company wants to create an chart to monitor the weight of the candy bars. Suppose 5 samples are taken, each of size 30, having the sample means 36.1, 34.6, 35.8, 34.9, and 35.8 grams, respectively. -Refer to Weights of Chocolate Bars Narrative. What is the centreline value?

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

The standard error of the sampling distribution of the sample proportion, , when n = 100 and p = 0.15 is 0.001275.

Weights of Chocolate Bars Narrative A candy bar factory is pouring chocolate into moulds to cool. The finished bars are sold as 35 gram bars. The company will lose money if the moulds are overfilled. If the moulds are underfilled, the weight of the candy bars will be less than the wrapper label says, and the company will be subject to a fine for misrepresenting the size of its product. The company wants to create an chart to monitor the weight of the candy bars. Suppose 5 samples are taken, each of size 30, having the sample means 36.1, 34.6, 35.8, 34.9, and 35.8 grams, respectively. -Refer to Weights of Chocolate Bars Narrative. The calculated value of s, the sample standard deviation of all nk = (30)(5) = 150 observations, is 3.36 grams. What is the standard error of the mean of 30 observations?

The sampling distribution of the sample mean, , is the distribution of all possible sample means that could be computed from all possible samples of a given size, n.

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

MRI Accuracy Narrative 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. -Refer to MRI Accuracy Narrative. Can valid inferences be made using the results of this study? Why or why not?

Canadian Cancer Society Survey Narrative A recent nationwide survey by the Canadian Cancer Society found the percentage of women who smoke has increased to 30%. That seemed a little low for your province, so you sampled 500 women from your province and found that 180 of them smoke. -Refer to Canadian Cancer Society Survey Narrative. Find the sample proportion of women who smoke in your province.

If a sample of n elements is selected from a population of N elements using a sampling plan in which each of the possible samples has the same chance of being selected, then the sampling is said to be random and the resulting sample is a simple random sample.

Miller Analogies Test Narrative Graduate students applying for entrance to many universities must take a Miller Analogies Test. It is known that the test scores have a mean of 75 and a variance of 16. In 1990, 100 students applied for entrance into graduate school in physics. -Refer to Miller Analogies Test Narrative. Find the mean and standard deviation of the sampling distribution of .

Defective Engine Parts Narrative A machine that manufactures a part for a car engine was observed over a period of time before a random sample of 300 parts was selected from those produced by this machine. Of the 300 parts, 15 were defective. -Refer to Defective Engine Parts Narrative. Completely describe the sampling distribution of , the proportion of defectives.

According to the Central Limit Theorem, if is the mean of a simple random sample taken from a large population, and if the N population values are normally distributed, the sampling distribution of is also normally distributed, regardless of sample size, n.

Rh-Positive Blood Type Narrative The proportion of individuals with an Rh-positive blood type is 88%. You have a random sample of n = 500 individuals. -Refer to Rh-Positive Blood Type Narrative. What are the mean and standard deviation of , the sample proportion with Rh-positive blood type?

The mean of the sample means and the standard deviation of 50 samples of size 5 taken from a production process under control are found to be 300 and 25, respectively. At which of the following values would the lower control limit for the chart be located?

A systematic random sample is a subset of a population, chosen by randomly selecting one of the first k elements and then including every kth element thereafter until the desired sample size has been reached.