Exam 7: Sampling Distributions

Unemployment Rate Narrative An eastern province has an unemployment rate of 6%. The province conducts monthly surveys in order to track the unemployment rate. In a recent month, a random sample of 700 people showed that 35 were unemployed. -Refer to Unemployment Rate Narrative. If Based on the most recent sample, find the probability that the sample proportion will lie within 0.005 of the true proportion p of people who are unemployed.

An in-control process is typically defined as a process in which all output is operating within 3 standard deviations of the centreline of the process.

If a sample is selected at random, the main reason that the sample mean, , might be different than the corresponding population mean, , is that the sample might be biased.

A systematic random sample might be used by a quality inspector who wants to determine the percentage of defective units produced on an assembly line that never shuts down.

Defective Items Narrative Suppose a sample of 120 items is drawn from a population of manufactured products and the number of defective items is recorded. Prior experience has shown that the proportion of defectives is 0.05. -Refer to Defective Items Narrative. Explain how to construct a p chart for the process and how it can be used.

If all possible samples of size n are drawn from a large population with a mean of 20 and a standard deviation of 5, then for which of the following samples sizes would the standard error of the sample mean equal 1.0?

A sampling distribution is defined as a sample chosen in such a way that every possible subset of like size has an equal chance of being selected.

The Central Limit Theorem applies to the sampling distribution of sample proportion, , but not to the sampling distribution of sample mean, .

Childcare Amounts Paid Narrative The distribution of the amount spent for childcare in a city has a mean of $675 and a standard deviation of $80. A random sample of 64 families paying for childcare is selected. -Refer to Childcare Amounts Paid Narrative. Describe the sampling distribution of .

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.

When a sample is selected at random from a population, which of the following will probably be the case?

Defective Items Narrative Suppose a sample of 120 items is drawn from a population of manufactured products and the number of defective items is recorded. Prior experience has shown that the proportion of defectives is 0.05. -Refer to Defective Items Narrative. If the production process is shut down whenever a sample proportion of defectives exceeds 0.05, what is the likelihood of this happening?

If a population standard deviation is equal to 50, then the sampling distribution of the sample mean, , will have a standard deviation that is less than 50 for all samples of size n > 2.

Stress and Sweets Narrative According to one study, 50% of Canadians admit to overeating sweet foods when stressed. Suppose that the 50% figure is correct and that a random sample of n = 100 Canadians is selected. -Refer to Stress and Sweets Narrative. What is the probability that lies within the interval 0.39 to 0.59?

Salaries of Professors Narrative Suppose that college faculty with the rank of professor at four-year institutions earn an average of $72,500 per year with a standard deviation of $4,500. In an attempt to verify this salary level, a random sample of 60 professors was selected from a personnel database for all four-year institutions in Canada. -Refer to Salaries of Professors Narrative. If your random sample actually produced a sample mean of $75,000, would you consider this unusual? What conclusion might you draw?

According to the Central Limit Theorem, for large samples, the standard error of the sample mean is the population standard deviation divided by the square root of the sample size.

Department Store Audits Narrative An auditor is going to sample 64 charge accounts from the many active accounts at a large department store. Suppose the average balance owed among all the store's accounts is $75 with a standard deviation of $100. -Refer to Department Store Audits Narrative. What is the probability the sample mean of the balances owed among the 64 accounts sampled by the auditor is below $70?

Federal Budget Narrative Suppose public opinion is split 65% against and 35% for increasing taxes to help balance the federal budget. Suppose also that 500 people from the population are selected randomly and interviewed. -Refer to Federal Budget Narrative. What is the probability the proportion favouring a tax increase is more than 30%?

In order to use the Central Limit Theorem to describe the sampling distribution of the sample mean, , the sample size, n, must be 30 or more.

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