Homework Week 6
Be advised, I may change some questions from the questions in the book. Please use this document as the source for the question for all homework.
1. Requests to a Web server
According to Brighton Webs LTD, a British company that specializes in data analysis, the arrival time of requests to a Web server within each hour can be modeled by a uniform distribution (www.brighton-webs.co.uk). Specifically, the number of seconds from the start of the hour that the request is made is uniformly distributed between 0 and 3,600 seconds. Find the probability that a request is made to a Web server sometime during the last 15 minutes of the hour.
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2. 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’s shots follows a normal distribution with a mean of zero feet and a standard deviation of 3 feet. (For example, if the player hits his target, x = 0 ; if he misses his target by 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 on both sides; 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 goalpost and the outer limit of the goalkeeper’s reach, what is the probability that he will score?
3. Industrial filling process
The characteristics of an industrial filling process in which an expensive liquid is injected into a container were investigated in the Journal of Quality Technology (July 1999). The quantity injected per container is approximately normally distributed with mean 10 units and standard deviation .2 unit. Each unit of fill costs $20. If a container contains less than 10 units (i.e., is underfilled), it must be reprocessed at a cost of $10. A properly filled container sells for $230.
a. Find the probability that a container is underfilled.
b. A container is initially underfilled and must be reprocessed. Upon refilling, it contains 10.6 units. How much profit will the company make on this container?
The operations manager adjusts the mean of the filling process upward to 10.5 units.
c. What is the probability of underfilling now?
d. Assuming Under these conditions, what is the expected profit per container?
4. Body fat in men
The percentage of fat in the bodies of American men is an approximately normal random variable with mean equal to 15% and standard deviation equal to 2%.
a. If these values were used to describe the body fat of men in the U.S. Army, and if a measure of 20% or more body fat characterizes the person as obese, what is the probability that a man in the US is obese?
b. What is the approximate probability that a random sample of 10,000 soldiers will contain fewer than 50 who would actually be characterized as obese? Use a binomial random variable for the number of men in the Army who are obese. Check that a normal distribution can be used as an approximation for the binomial distribution.
c. If the army actually were to check the percentage of body fat for a random sample of 10,000 men, and if only 30 contained 20% (or higher) body fat, would you conclude that the army was successful in reducing the percentage of obese men below the percentage in the general population? Explain your reasoning
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