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Instructions for a project in stt 315 where students are required to use microsoft excel to compute binomial and hypergeometric probabilities. The project involves answering questions related to binomial and hypergeometric distributions using excel functions. The document also includes instructions on how to use excel to compute binomial probabilities and hypergeometric probabilities.
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STT 315: Project 1 Instructions and explanation Fall 2002
Microsoft Excel has several probability distributions available as functions, including the binomial and hypergeo- metric distributions. We’ll investigate how to compute binomial probabilities in Excel in the context of a simple example.
Example 1. A firm plans to contact businesses in a large city by mail as part of a survey. Surveys will be sent by mail to a random sample of n = 200 businesses in the area. From past surveys, the firm expects that 35% of the businesses will respond to the survey. Let X represent the number of the 200 businesses that respond. We will model X by a binomial distribution with n = 200 and p = 0.35.
(a) What is the probability that exactly 72 businesses respond?
(b) What is the probability that the response rate is at most 0.30?
(c) What is the probability that more than 70 businesses respond?
1.2.1 Part (a)
Part (a) asks for P (X = 72). Here’s how compute this using Excel.
1.2.2 Part (b)
For Part (b) we need to do a bit of work setting up the problem before Excel can help. Since the response rate is X/200, we want P (X/ 200 ≤ 0 .30).
Multiply both sides by 200 to rewrite this in terms of X:
P (X ≤ 60).
So we want the value of the cumulative distribution function at 60. To obtain this from Excel, repeat the steps above, with the following changes:
Your answer should be 0.0783.
1.2.3 Part (c)
For Part (c) we want P (X > 70). This is equal to 1−P (X ≤ 70) by the complement rule. So compute P (X ≤ 70) in Excel, and then subtract the answer from one, either by hand or using Excel. Your answer should be 0.46753.
Example 2. A shipment of 350 parts contains 22 defective parts. An inspector chooses 40 parts at random to be inspected.
(a) What is the probability that exactly 2 of the inspected parts are defective?
(b) What is the probability that more than 5 of the inspected parts are defective?
1.3.1 Part (a)
To answer Part (a), we need to compute P (X = 2), where X represents the number of defective parts in the 40 parts inspected. We know that X has a hypergeometric distribution with N = 350, S = 22, n = 40. Here’s how to compute the probability using Excel.
For some values of n and p, when n is rather large (a bit over 1000), Excel has trouble computing the appropriate binomial probabilities. Similarly, when N is large (again, a bit over 1000), Excel has trouble computing the appropriate hypergeometric probabilities. Unfortunately,
Example 3. Let X have a binomial distribution with parameters n = 1030 and p = 0.5. We will try to compute P (X = 498), P (X = 499), P (X = 500), and P (X = 501) using Excel.
The example above, and other examples where Excel does not do probability and statistical calculations properly, can be found in a paper by Leo Kn¨usel of the University of Munich. The paper and related papers are available on the web at http://www.stat.uni-muenchen.de/~knusel/elv/accuracy.html.