Standard Deviation

1. Suppose that in a very large supply of compact discs the mean bad-sector volume is 3.5 K (kilobytes) per disc and standard deviation .83 K. A retailer assembles packs of 100 compact discs from the supply.

(a) Let X¯ be the average bad-sector volume per disc in a randomly selected pack. What is the expected value of X¯?

(b) What is the standard deviation of X¯?

(c) What is the (approximative) probability that X¯ is below 3.6 K?

2. A university must decide between two health plans for its faculty and stuff. An important piece of information in making this decision is the average annual medical expenses of the individual employees. A random sample of 20 employees was surveyed. In this group the average annual medical expense was $1,982. The sample standard deviation is s = $360. Construct a 95% confidence interval for the true average medical expense of the university employees. Assume that the population distribution is normal.

3. An agency wants to estimate the fraction of votes candidate A will receive in an election. The objective is that with 95% probability the estimate and the actual fraction of votes candidate A will receive do not differ more than .03.

(a) What sample size should the agency select in the poll to satisfy this requirement if they have no information concerning the fraction of votes candidate A will receive?

(b) Suppose now that we know that candidate A will receive at least 80% of the votes. What sample size is necessary to achieve the same objective as in part (a)?

(c) Suppose that this is a particularly tight election. In particular, you know that the fraction of votes candidate A will receive will be between 45% and 54%. What sample size is necessary to achieve the same objective as in part (a)?

4. An automobile insurance firm wants to find the average amount per claim for auto body repairs. Its summary records combine amounts for body repair with all other repair amounts, so a sample of individual claims must be taken. Their goal is to estimate the average claim for auto body repairs with a precision of $35, with 95% confidence. In other words, they want to be 95% confident that the difference between the population mean and the sample mean is no more than $35. A “horseback guess” says that the standard deviation is about $400. How large a sample is needed?

5. In a marketing study 100 customers have been selected randomly from a large population. In this sample, 60 people said that they prefer brand A against brand B. Construct a 95% 1 2 confidence interval for the proportion of people within the population who prefer brand A against brand B.

6. Company A receives regular delivery of apples from a vendor. The mean weight of these apples is 150 grams, and the standard deviation is 20 grams. These apples arrive in boxes, each box containing 100 apples. Upon receiving a shipment, the quality control department of company A measures the net weight of each box. Boxes with total net weight less than 14.8 kg (1kg=1,000 grams) will be returned to the vendor.

(a) What is the probability that a box will be returned to the vendor from company A?

(b) Boxes that are returned to the vendor are immediately sent to company B. The quality control department at company B will return boxes with total net weight below 14.6 kg. What is the probability that a box recived by company B will be returned to the vendor?

(c) What fraction of all boxes will be returned by both company A and company B?