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Chapter 8: Q44E (page 452)
For each of the following values of find the values of z for which would be rejected in favor of .
a. For ,the value of Zis -2.326.
b. For ,the value of Zis -1.96.
c. For ,the value of Zis -1.645.
d. For ,the value of Z is -1.28.
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Optimal goal target in soccer. When attempting to score a goal in soccer, where should you aim your shot? Should you aim for a goalpost (as some soccer coaches teach), the middle of the goal, or some other target? To answer these questions, Chance (Fall 2009) utilized the normal probability distribution. Suppose the accuracy x of a professional soccer player鈥檚 shots follows a normal distribution with a mean of 0 feet and a standard deviation of 3 feet. (For example, if the player hits his target,x=0; if he misses his target 2 feet to the right, x=2; and if he misses 1 foot to the left,x=-1.) Now, a regulation soccer goal is 24 feet wide. Assume that a goalkeeper will stop (save) all shots within 9 feet of where he is standing; all other shots on goal will score. Consider a goalkeeper who stands in the middle of the goal.
a. If the player aims for the right goalpost, what is the probability that he will score?
b. If the player aims for the center of the goal, what is the probability that he will score?
c. If the player aims for halfway between the right goal post and the outer limit of the goalkeeper鈥檚 reach, what is the probability that he will score?
In a random sample of 250 people from a city, 148 of them favor apples over other fruits.
a. Use a 90% confidence interval to estimate the true proportion p of people in the population who favor apples over other fruits.
b. How large a sample would be needed to estimate p to be within .15 with 90% confidence?
Gouges on a spindle. A tool-and-die machine shop produces extremely high-tolerance spindles. The spindles are 18-inch slender rods used in a variety of military equipment. A piece of equipment used in the manufacture of the spindles malfunctions on occasion and places a single gouge somewhere on the spindle. However, if the spindle can be cut so that it has 14 consecutive inches without a gouge, then the spindle can be salvaged for other purposes. Assuming that the location of the gouge along the spindle is random, what is the probability that a defective spindle can be salvaged?
Question: The purpose of this exercise is to compare the variability of with the variability of .
a. Suppose the first sample is selected from a population with mean and variance . Within what range should the sample mean vary about of the time in repeated samples of measurements from this distribution? That is, construct an interval extending standard deviations of on each side of .
b. Suppose the second sample is selected independently of the first from a second population with mean and variance . Within what range should the sample mean vary about the time in repeated samples of measurements from this distribution? That is, construct an interval extending standard deviations on each side .
c. Now consider the difference between the two sample means . What are the mean and standard deviation of the sampling distribution ?
d. Within what range should the difference in sample means vary about the time in repeated independent samples of measurements each from the two populations?
e. What, in general, can be said about the variability of the difference between independent sample means relative to the variability of the individual sample means?
Product failure behavior. An article in Hotwire (December 2002) discussed the length of time till the failure of a product produced at Hewlett Packard. At the end of the product鈥檚 lifetime, the time till failure is modeled using an exponential distribution with a mean of 500 thousand hours. In reliability jargon, this is known as the 鈥渨ear-out鈥� distribution for the product. During its normal (useful) life, assume the product鈥檚 time till failure is uniformly distributed over the range of 100 thousand to 1 million hours.
a. At the end of the product鈥檚 lifetime, find the probability that the product fails before 700 thousand hours.
b. During its normal (useful) life, find the probability that the product fails before 700 thousand hours.
c. Show that the probability of the product failing before 830 thousand hours is approximately the same for both the normal (useful) life distribution and the wear-out distribution.
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