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Chapter 5: Problem 73

York Steel Corporation produces a special bearing that must meet rigid specifications. When the production process is running properly, \(10 \%\) of the bearings fail to meet the required specifications. Sometimes problems develop with the production process that cause the rejection rate to exceed \(10 \%\). To guard against this higher rejection rate, samples of 15 bearings are taken periodically and carefully inspected. If more than 2 bearings in a sample of 15 fail to meet the required specifications, production is suspended for necessary adjustments. a. If the true rate of rejection is \(10 \%\) (that is, the production process is working properly), what is the probability that the production will be suspended based on a sample of 15 bearings? b. What assumptions did you make in part a?

Short Answer

Expert verified
a. The probability that the production will be suspended based on a sample of 15 bearings, given a true rejection rate of 10 percent, is 1 minus the cumulative binomial probability up to 2 defective bearings. b. The assumptions made are that each bearing is an independent trial, each trial only has two outcomes (defective or not defective), and the probability of getting a defective bearing remains constant for each trial.

Step by step solution

01

Identify Variables and Parameters

The number of trials (n) is 15 (number of bearings in a sample), number of failures (x) that will trigger the production suspension is greater than 2 and the success probability (p) is 0.1 (10% rate of rejection).
02

Use the Cumulative Density Function (CDF)

To calculate the probability that production will be suspended (i.e., more than 2 bearings in a sample of 15 are defective), we use the cumulative density function of binomial distribution which gives the probability of getting a certain number of 'successes' or less. We need to compute the cumulative probability up to 2 defective bearings and subtract it from 1 to get the probability of more than 2 defective bearings.
03

Compute Cumulative Binomial Probability

The cumulative binomial probability can be computed using the formula:\[F(k; n, p) = \sum_{i=0}^{k} C(n, i) \times p^i \times (1-p)^{n-i}\] Where \(C(n, i)\) is the combination 'n choose i', n is total number of trials, i is the number of 'successes', 'p' is the probability of success
04

Subtract the Cumulative Probability from 1

To find the probability of getting more than 2 defective bearings, we compute the cumulative probability up to 2 defective bearings and subtract it from 1.

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Most popular questions from this chapter

Which of the following are binomial experiments? Explain why. a. Rolling a die 10 times and observing the number of spots b. Rolling a die 12 times and observing whether the number obtained is even or odd c. Selecting a few voters from a very large population of voters and observing whether or not each of them favors a certain proposition in an election when \(54 \%\) of all voters are known to be in favor of this proposition.

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