91Ó°ÊÓ

Does a raw score less than the mean correspond to a positive or negative standard score? What about a raw score greater than the mean?

What is a population parameter? Give three examples.

Empirical Rule What percentage of the area under the normal curve lies (a) to the left of \(\mu ?\) (b) between \(\mu-\sigma\) and \(\mu+\sigma ?\) (c) between \(\mu-3 \sigma\) and \(\mu+3 \sigma ?\)

Insurance: Claims Do you try to pad an insurance claim to cover your deductible? About \(40 \%\) of all U.S. adults will try to pad their insurance claims! (Source: Are You Normal?, by Bernice Kanner, St. Martin's Press.) Suppose that you are the director of an insurance adjustment office. Your office has just received 128 insurance claims to be processed in the next few days. What is the probability that (a) half or more of the claims have been padded? (b) fewer than 45 of the claims have been padded? (c) from 40 to 64 of the claims have been padded? (d) more than 80 of the claims have not been padded?

Coal is carried from a mine in West Virginia to a power plant in New York in hopper cars on a long train. The automatic hopper car loader is set to put 75 tons of coal into each car. The actual weights of coal loaded into each car are normally distributed, with mean \(\mu=75\) tons and standard deviation \(\sigma=0.8\) ton. (a) What is the probability that one car chosen at random will have less than 74.5 tons of coal? (b) What is the probability that 20 cars chosen at random will have a mean load weight \(\bar{x}\) of less than 74.5 tons of coal? (c) Interpretation Suppose the weight of coal in one car was less than 74.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment? Suppose the weight of coal in 20 cars sclected at random had an average \(\bar{x}\) of less than 74.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment? Why?

Find the \(z\) value described and sketch the area described.Find \(z\) such that \(97.5 \%\) of the standard normal curve lies to the left of \(z\).

Sketch the areas under the standard normal curve over the indicated intervals and find the specified areas. To the left of \(z=0.72\)

Let \(x\) represent the dollar amount spent on supermarket impulse buying in a 10 -minute (unplanned) shopping interval. Based on a Denver Post article, the mean of the \(x\) distribution is about \(\$ 20\) and the estimated standard deviation is about \(\$ 7\) (a) Consider a random sample of \(n=100\) customers, each of whom has 10 minutes of unplanned shopping time in a supermarket. From the central limit theorem, what can you say about the probability distribution of \(\bar{x}\) the average amount spent by these customers due to impulse buying? What are the mean and standard deviation of the \(\bar{x}\) distribution? Is it necessary to make any assumption about the \(x\) distribution? Explain. (b) What is the probability that \(\bar{x}\) is between \(\$ 18\) and \(\$ 22 ?\) (c) Let us assume that \(x\) has a distribution that is approximately normal. What is the probability that \(x\) is between \(\$ 18\) and \(\$ 22 ?\) (d) Interpretation: In part (b), we used \(\bar{x},\) the average amount spent, computed for 100 customers. In part (c), we used \(x,\) the amount spent by only one customer. The answers to parts (b) and (c) are very different. Why would this happen? In this example, \(\bar{x}\) is a much more predictable or reliable statistic than \(x\). Consider that almost all marketing strategies and sales pitches are designed for the average customer and not the individual customer. How does the central limit theorem tell us that the average customer is much more predictable than the individual customer?

Is \(\hat{p}\) an unbiased estimator for \(p\) when \(n p>5\) and \(n q>5 ?\) Recall that a statistic is an unbiased estimator of the corresponding parameter if the mean of the sampling distribution equals the parameter in question.

Find the \(z\) value described and sketch the area described.Find the \(z\) value such that \(98 \%\) of the standard normal curve lies between \(-z\) and \(z\).