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The restaurant in a large commercial building provides coffee for the occupants in the building. The restaurateur has determined that the mean number of cups of coffee consumed in a day by all the occupants is 2.0 with a standard deviation of .6. A new tenant of the building intends to have a total of Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 313 SAMPLING DISTRIBUTIONS 9-2 SAMPLING DISTRIBUTION OF A PROPORTION In Section 7-4, we introduced the binomial distribution whose parameter is p, the probability of success in any trial. In order to compute binomial probabilities, we assumed that p was known. However, in the real world p is unknown, requiring the statistics practitioner to estimate its value from a sample. The estimator of a population proportion of successes is the sample proportion; that is, we count the number of successes in a sample and compute P ^ = X n (P ^ is read as p hat ) where X is the number of successes and n is the sample size. When we take a sample of size n, we’re actually conducting a binomial experiment; as a result, X is binomially distributed. Thus, the probability of any value of P ^ can be calculated from its value of X . For example, suppose that we have a binomial experiment with n = 10 and p = .4. To find the probability that the sample proportion P ^ is less than or equal to .50, we find the probability that X is less than or equal to 5 (because 5/10 = .50). From Table 1 in Appendix B we find with n = 10 and p = .4 P(P ^ = .50) = P(X = 5) = .8338 We can calculate the probability associated with other values of P ^ similarly. Discrete distributions such as the binomial do not lend themselves easily to the kinds of calculation needed for inference. And inference is the reason we need sampling distributions. Fortunately, we can approximate the binomial distribution by a normal distribution. What follows is an explanation of how and why the normal distribution can be used to approximate a binomial distribution. Disinterested readers can skip to page 317, where we present the approximate sampling distribution of a sample proportion. 9-2a (Optional) Normal Approximation to the Binomial Distribution Recall how we introduced continuous probability distributions in Chapter 8. We developed the density function by converting a histogram so that the total area in the rectangles equaled 1. We can do the same for a binomial distribution. To illustrate, let X be a binomial random variable with n = 20 and p = .5. We can easily determine the probability of each value of X , where X = 0, 1, 2, · · · , 19, 20. A rectangle representing a value of x is drawn so that its area equals the probability. We accomplish this by letting the height of the rectangle equal the probability and the base of the rectangle equal 1. Thus, the base of each rectangle for x is the interval x - .5 to x + .5. Figure 9.7 depicts this graph. As you can see, the rectangle representing x = 10 is the rectangle whose base is the interval 9.5 to 10.5 and whose height is P(X = 10) = .1762. 125 new employees. What is the probability that the new employees will consume more than 240 cups per day? 9.29 The number of pages produced by a fax machine in a busy office is normally distributed with a mean of 275 and a standard deviation of 75. Determine the probability that in 1 week (5 days) more than 1,500 faxes will be received?
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