Exam 7: Sampling Distributions

Suppose the monthly rents of all one-bedroom apartments in a small town are known to be normally distributed with a mean equal to $175 a month and standard deviation of $35. Which of the following would be the highest individual rent that you might expect to find?

Numerical descriptive measures calculated from a sample are called statistics.

Weights of Candy Bars Narrative A candy bar company is interested in reducing the percentage of defective candy bars made, where a defective candy bar is one that has too few almonds by weight. The company randomly samples 100 candy bars a day for 5 days and finds the percentage of the defective bars to be 0.0200, 0.0125, 0.0225, 0.0100, and 0.0150, respectively. The company wants to construct a control chart for the proportion defective in samples of size n = 100. -Refer to Weights of Candy Bars Narrative. What are the control limits?

A statistics professor has stated that 90% of his students pass the class. To check this claim, a random sample of 150 students indicated that 129 passed the class. If the professor's claim is correct, what is the probability that 129 or fewer will pass the class this semester?

Fifty samples of size 500 were drawn from a manufacturing process and the number of defectives in each sample was counted. The average of the sample proportion was 0.032. At which of the following values would the centreline for the p chart be located?

Content Composition Analysis Narrative A well-known juice manufacturer claims that its citrus punch contains 15% real orange juice. A random sample of 150 cans of the citrus punch is selected and analyzed for content composition. -Refer to Content Composition Analysis Narrative. Find the probability that the sample proportion will be less than 0.10.

A systematic random sample is a subset of a population chosen by taking separate censuses in a randomly chosen subset of distinct clusters into which the population is naturally divided.

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. Describe the sampling distribution of the proportion of defectives for a simple random sample of n = 50.

Which of the following does NOT correctly describe a random sample?

The standard error of the sampling distribution of sample proportion, SE( ), depends on the value of the population proportion, p, and the closer the value of p to 0.50, the larger SE( ) will be for a given sample size n.

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. Calculate the probability that the sample mean is greater than $75,000.

Random variation in a process can be eliminated.

The Central Limit Theorem states that the sample mean, , is always equal to the population mean, .

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. Is the normal approximation to the sampling distribution of appropriate in this situation? Explain.

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. Describe the sampling distribution and comment.

Deli Sales Narrative The total daily sales, x, in the deli section of a large chain of food stores is the sum of the sales generated by a fixed number of customers who make purchases on a given day. -Refer to Deli Sales Narrative. For this particular market, the average sale per customer in the deli section is $10.50 with = $2.50. If 30 customers make deli purchases on a given day, give the mean and standard deviation of the probability distribution of the total sales, x.

According to the Central Limit Theorem, any sampling distribution of is considered normal, provided .

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. Find the proportion of defective parts in the sample.

Lifetime of Fluorescent Light Bulbs The mean lifetime of a fluorescent light bulb is 1570 hours with a standard deviation of 200 hours. Suppose we take a sample of 100 bulbs, test them, and calculate the sample mean. -Refer to Lifetime of Fluorescent Light Bulbs. What is the probability the sample mean exceeds 1560 hours?

The most important contribution of the Central Limit Theorem is in statistical inference.