Question

A machine is shut down for repairs if a random sample of 100 items selected from the daily output of the machine reveals at least $15 \%$ defectives. (Assume that the daily output is a large number of items.) If on a given day the machine is producing only $10 \%$ defective items, what is the probability that it will be shut down? [Hint: Use the. 5 continuity correction.]

Name: Problem 72, Chapter 7: Mathematical Statistics with Applications
Uploaded: 2022-09-19T20:04:49-08:00
Channel: Jason Gerber
Description: A machine is shut down for repairs if a random sample of 100 items selected from the daily output of the machine reveals at least 15 % defectives. (Assume that the daily output is a large number of items.) If on a given day the machine is producing only 10 % defective items, what is the probability that it will be shut down? [Hint: Use the. 5 continuity correction.]

   A machine is shut down for repairs if a random sample of 100 items selected from the daily  output of the machine reveals at least $15 \%$ defectives. (Assume that the daily output is a large number of items.) If on a given day the machine is producing only $10 \%$ defective items, what is the probability that it will be shut down? [Hint: Use the. 5 continuity correction.]

Mathematical Statistics with Applications

Dennis D. Wackerly,… 7th Edition

Chapter 7, Problem 72

Instant Answer

Solved by Expert Jason Gerber

09/19/2022

Step 1

The number of trials, n, is 100. The probability of success, p, is 10% or 0.1. Therefore, the probability of failure, 1-p, is 0.9. Show more…

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A machine is shut down for repairs if a random sample of 100 items selected from the daily output of the machine reveals at least $15 \%$ defectives. (Assume that the daily output is a large number of items.) If on a given day the machine is producing only $10 \%$ defective items, what is the probability that it will be shut down? [Hint: Use the. 5 continuity correction.]

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Key Concepts

Binomial Distribution

This concept involves discrete probability distributions used to model the number of successes in a fixed number of independent trials, each with the same probability of success. It is particularly applicable in quality control scenarios where items are classified as defective or non-defective.

Normal Approximation

When dealing with large sample sizes, the binomial distribution can be approximated by a normal distribution according to the Central Limit Theorem. This approximation simplifies the calculation of probabilities for discrete outcomes by using a continuous probability distribution.

Continuity Correction

When approximating a discrete distribution (such as the binomial) with a continuous one (like the normal), the continuity correction adjustment (adding or subtracting 0.5) is used. This correction improves the accuracy of the approximation by accounting for the discrete nature of the original variable.

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