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Chapter 3: Q100E (page 137)

A manufacturer of integrated circuit chips wishes to control the quality of its product by rejecting any batch in which the proportion of defective chips is too high. To this end, out of each batch (10,000 chips), 25 will be selected and tested. If at least 5 of these 25 are defective, the entire batch will be rejected.

a. What is the probability that a batch will be rejected if 5% of the chips in the batch are in fact defective?

b. Answer the question posed in (a) if the percentage of defective chips in the batch is \({\bf{10}}\% \).

c. Answer the question posed in (a) if the percentage of defective chips in the batch is \({\bf{20}}\% \).

d. What happens to the probabilities in (a)鈥�(c) if the critical rejection number is increased from 5 to \({\bf{6}}\)?

Short Answer

Expert verified

(a) The probability is \(P(X \ge 5) = 0.007\).

(b) The percentage of defective chips in the batch is \(10\% \)is \(P(X \ge 5) = 0.098\).

(c) The percentage of defective chips in the batch is \(20\% \)is \(P(X \ge 5) = 0.579\).

(d) If the critical rejection number is increased from 5 to \(6\), So The probabilities would decrease.

Step by step solution

01

Concept Introduction

Probability is the likelihood that an event will occur and is calculated by dividing the number of favourable outcomes by the total number of possible outcomes. The simplest example is a coin flip. When you flip a coin there are only two possible outcomes, the result is either heads or tails.

02

Determine the probability

(a)

The second parameter of the specified Binomial Distribution is \({\bf{p}}\)in this example

\(p = 0.05\)

If at least \(5\) out of \(25\) chips are damaged, the entire batch will be rejected; thus, the requested probability is

\(P(X \ge 5) = 1 - B(4;25,0.05) = 1 - 0.993\)

\({\rm{ = 0}}{\rm{.007}}\)

Hence, the probability is \(P(X \ge 5) = 0.007\).

03

Determine the percentage of defective chips

(b)

The second parameter of the specified Binomial Distribution is \(p\)in this example.

\(p = 0.1\)

If at least 5 out of 25 chips are damaged, the entire batch will be rejected; thus, the requested probability is

\(P(X \ge 5) = {\rm{ }}1 - B(4;25,0.1) = 1 - 0.902\)

=0.098

Therefore, the percentage of defective chips in the batch is \(10\% \)is \(P(X \ge 5) = 0.098\).

04

Determine the percentage of defective chips

(c)

The second parameter of the specified Binomial Distribution is \({\bf{p}}\)in this example.

\(p = 0.2\)

If at least \(5\) out of \(25\) chips are damaged, the entire batch will be rejected; thus, the requested probability is

\(P(X \ge 5) = 1 - B(4;25,0.2) = 1 - 0.421\)

\({\rm{ = 0}}{\rm{.579}}\)

Thus, the percentage of defective chips in the batch is \(20\% \)is \(P(X \ge 5) = 0.579\).

05

Determine the probability

(d)

The possibilities would dwindle (look at the Cumulative Density Function - sum of positive numbers).

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Most popular questions from this chapter

Consider writing onto a computer disk and then sending it through a certifier that counts the number of missing pulses. Suppose this number X has a Poisson distribution with parameter\({\rm{\mu = }}{\rm{.2}}\). (Suggested in 鈥淎verage Sample Number for Semi-Curtailed Sampling Using the Poisson Distribution,鈥� J. Quality Technology,\({\rm{1983 = 126 - 129}}\).) a. What is the probability that a disk has exactly one missing pulse? b. What is the probability that a disk has at least two missing pulses? c. If twodisks are independently selected, what is the probability that neither contains a missing pulse?

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a. What is the \({\rm{pmf}}\)of the number of granite specimens selected for analysis?

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Each of \({\rm{12}}\)refrigerators of a certain type has been returned to a distributor because of an audible, high-pitched, oscillating noise when the refrigerators are running. Suppose that \({\rm{7}}\) of these refrigerators have a defective compressor and the other \({\rm{5}}\) have less serious problems. If the refrigerators are examined in random order, let\({\rm{X}}\)be the number among the first \({\rm{6}}\) examined that have a defective compressor.

a. Calculate\({\rm{P(X = 4)}}\)and \(P(X拢 4)\)

b. Determine the probability that \({\rm{X}}\) exceeds its mean value by more than \({\rm{1}}\) standard deviation.

c. Consider a large shipment of \({\rm{400}}\)\({\rm{40}}\) refrigerators, of which 40 have defective compressors. If \({\rm{X}}\) is the number among \({\rm{15}}\) randomly selected refrigerators that have defective compressors, describe a less tedious way to calculate (at least approximately) P(X拢5)than to use the hypergeometric \({\rm{pmf}}\).

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Customers at a gas station pay with a credit card (A), debit card (B), or cash (C). Assume that successive customers make independent choices, with P(A)\({\rm{ = }}{\rm{.5}}\), P(B)\({\rm{ = }}{\rm{.2}}\), and P(C)\({\rm{ = }}{\rm{.3}}\). a. Among the next\({\rm{100}}\)customers, what are the mean and variance of the number who pay with a debit card? Explain your reasoning. b. Answer part (a) for the number among the\({\rm{100}}\)who don鈥檛 pay with cash.
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