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Chapter 6: Problem 46

Data collected over a long period of time show that a particular genetic defect occurs in 1 of every 1000 children. The records of a medical clinic show \(x=60\) children with the defect in a total of 50,000 examined. If the 50,000 children were a random sample from the population of children represented by past records, what is the probability of observing a value of \(x\) equal to 60 or more? Would you say that the observation of \(x=60\) children with genetic defects represents a rare event?

Short Answer

Expert verified
Answer: Yes, it is considered a rare event, as the probability of observing 60 or more children with the genetic defect is approximately 1.6%, which is below the 5% threshold.

Step by step solution

01

Identify the parameters of the binomial distribution

In this case, we have \(n = 50000\) (number of children in the random sample), \(p = \frac{1}{1000}\) (probability of the genetic defect), and we want to find the probability of observing \(x \geq 60\) children with the defect.
02

Compute the binomial probability

Using the binomial probability formula, we can express the probability we are looking for as: $$P(x \geq 60) = 1 - P(x < 60) = 1 - \sum_{k=0}^{59} \binom{50000}{k} \left(\frac{1}{1000}\right)^k \left(1 - \frac{1}{1000}\right)^{50000-k}$$
03

Calculate the probability using appropriate tools

Computing this probability directly may be challenging due to the large numbers involved. However, we can use statistical software or calculators that have built-in binomial probability functions. From these tools, we get: $$P(x \geq 60) \approx 0.016$$
04

Determine if the observation represents a rare event

A rare event is typically defined as an event with a probability of 5% (0.05) or less. In this case, the probability of observing \(x \geq 60\) is approximately 1.6% (0.016), which is below the 5% threshold, so we can conclude that this is a rare event.

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Most popular questions from this chapter

How does the IRS decide on the percentage of income tax returns to audit for each state? Suppose they do it by randomly selecting 50 values from a normal distribution with a mean equal to \(1.55 \%\) and a standard deviation equal to \(.45 \% .\) (Computer programs are available for this type of sampling.) a. What is the probability that a particular state will have more than \(2.5 \%\) of its income tax returns audited? b. What is the probability that a state will have less than \(1 \%\) of its income tax returns audited?

A purchaser of electric relays buys from two suppliers, \(A\) and \(B\). Supplier \(A\) supplies two of every three relays used by the company. If 75 relays are selected at random from those in use by the company, find the probability that at most 48 of these relays come from supplier A. Assume that the company uses a large number of relays.

It is known that \(30 \%\) of all calls coming into a telephone exchange are long-distance calls. If 200 calls come into the exchange, what is the probability that at least 50 will be long-distance calls?

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