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Chapter 5: Problem 25

Each item produced by a certain manufacturer is, independently, of acceptable quality with probability.95. Approximate the probability that at most 10 of the next 150 items produced are unacceptable.

Short Answer

Expert verified
The probability that at most 10 of the next 150 items produced are unacceptable is approximately 0.874, using the normal distribution as an approximation for the given binomial distribution with n = 150 and p = 0.05.

Step by step solution

01

Identify the given information and required probability

We are given that an item is of acceptable quality with a probability of 0.95. Therefore, the probability that an item is unacceptable is 1 - 0.95 = 0.05. We need to find the probability that at most 10 of the next 150 items produced are unacceptable, which can be denoted as P(X 鈮� 10), where X follows a binomial distribution with parameters n = 150 and p = 0.05.
02

Approximate using the normal distribution

Since the binomial distribution has large n (n = 150), we can approximate it using the normal distribution. To do this, we need to find the mean (渭) and standard deviation (蟽) of the binomial distribution: 渭 = n * p = 150 * 0.05 = 7.5 蟽 = 鈭�(n * p * (1 - p)) = 鈭�(150 * 0.05 * 0.95) 鈮� 2.598 Now, let's use the normal distribution with mean 渭 = 7.5 and standard deviation 蟽 = 2.598 to approximate the probability P(X 鈮� 10).
03

Apply continuity correction and find the z-score

Before finding the probability, we need to apply the continuity correction. Since we want the probability that X 鈮� 10, we will use 10.5 as the upper boundary (as it includes the possibility of exactly 10). Now, let's find the z-score corresponding to 10.5: z = (x - 渭) / 蟽 = (10.5 - 7.5) / 2.598 鈮� 1.155
04

Find the probability using z-score

Now that we have the z-score, we can find the probability using the standard normal table or a calculator: P(X 鈮� 10) = P(Z 鈮� 1.155) 鈮� 0.874 Hence, the probability that at most 10 of the next 150 items produced are unacceptable is approximately 0.874.

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