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The write right company manufactures ballpoint pens and has been experiencing a 5% rate of defective pens. modifications are made to the manufacturing process in an attempt to improve quality, and the manager claims that the modified procedure is better, because a test of 50 pens shows that only one is defective. a. assuming that the 5% rate of defects has not changed, find the probability that among 50 pens, exactly one is defective. b. assuming that the 5% rate of defects has not changed, find the probability that among 50 pens, none are defective. c. what probability value should be used for determining whether the modified process results in a defect rate that is less than 5%? d. what do you conclude about the effectiveness of the modified production process?

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a. The probability that exactly one is defective is 0.2025

b. The probability that none will be defective is 0.0769

c. The probability rate that should be used is 2%

d. The modified production process is better.

The question has to do with binomial probability.

What is binomial probability?

Binomial probability is probability in which the event can only have two values or is binary.

Given the binomial probability formula,

P(X = x) = ⁿCₓpˣ(1 - p)ⁿ ⁻ ˣ

where

  • p = probability of defective = 5% = 0.05 and
  • n = 50

a. Assuming that the 5% rate of defects has not changed, find the probability that among 50 pens, exactly one is defective.

For the probability that exactly one is defective, x = 1

So, P(x = 1) = ⁵⁰C₁p¹(1 - p)⁵⁰ ⁻ ¹

= ⁵⁰C₁p(1 - p)⁴⁹

= ⁵⁰C₁(0.05)(1 - 0.05)⁴⁹

= ⁵⁰C₁(0.05)(0.95)⁴⁹

= 50(0.05)(0.95)⁴⁹

= 50(0.05)(0.081)

= 0.2025

So, the probability that exactly one is defective is 0.2025

b. Assuming that the 5% rate of defects has not changed, find the probability that among 50 pens, none are defective.

For the probability that exactly one is defective, x = 0

So, P(x = 0) = ⁵⁰C₀p⁰(1 - p)⁵⁰ ⁻ ⁰

= ⁵⁰C₀(1 - p)⁵⁰

= ⁵⁰C₀(1 - 0.05)⁵⁰

= ⁵⁰C₀(0.95)⁵⁰

= 1 × (0.95)⁵⁰

= 0.0769

So, the probability that none will be defective is 0.0769

c. What probability value should be used for determining whether the modified process results in a defect rate that is less than 5%?

Since the modified result results in a defective rate of one out of 50 pens.

The probability rate that should be used is p = number of defective pens/total number of pens = 1/50

= 1/50 × 100 %

= 2%

The probability rate that should be used is 2%

d. What do you conclude about the effectiveness of the modified production?

Since the defective rate of the modified production process is 2% which is less than that of the previous production process which is 5%, the modified production process is better.

So, the modified production process is better.

Learn more about binomial probability here:

https://brainly.com/question/13018489

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