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An electronic component which forms part of an LED system is mass produced. A series of tests over a long period of time has revealed that on average 4% of these components are defective. The production company proposes two possible schemes, described as Sampling methods A and B below, for monitoring the quality of a large batch of these components, by taking and testing small samples.

Sampling method (A): A single sample of 15 components is drawn at random from a large batch and inspected for defective components. If this sample contains 0 or 1 defectives then the batch is accepted for selling, otherwise it is rejected.

(i)     Use the Binomial distribution to calculate, correct to 4 decimal places, the probability     that the batch is accepted.


Sampling method (B): A first sample of 20 components is drawn from the batch and inspected for defective components. If this first sample contains fewer than 3 defectives then the batch is accepted. If the first sample contains more than 3 defectives then the batch is rejected. If the first sample contains exactly 3 defectives then a second sample, of size 10, is taken. When taken, if this second sample contains fewer than 2 defectives, then the batch is accepted, otherwise it is rejected.

In this case, a compound statement which defines the acceptance of the batch is as follows: [Sample 1 contains fewer than 3 defectives] OR [(Sample 1 contains exactly 3 defectives) AND (Sample 2 contains fewer than 2 defectives)].

(ii)    For sampling method B, use the Binomial distribution to calculate, correct to 4 decimal places, the probability that the first sample contains fewer than 3 defectives, the probability that the first sample contains exactly 3 defectives and the probability that the second sample contains fewer than 2 defectives.


(iii)    Represent Sampling method (B) by means of a Decision Tree showing all outcomes and associated probabilities. Hence or otherwise calculate, correct to 4 decimal places, the probability that a batch is accepted.


Sampling method A costs £30 to set up and Sampling method B costs £31 to set up. Both also incur an additional cost of 30p per component inspected.

(iv)    Calculate the cost of sampling using Sampling method (A).


The proportion of times that a second sample is taken in Sampling method (B) is equal to the probability that sample 1 has exactly 3 defectives.

(v)    Calculate, correct to the nearest penny, the expected (or average) cost of sampling according to Sampling method (B).


(vi)    With reference to your calculated probabilities of batch acceptance and costs of the two sampling plans, comment briefly on the business implications of implementing the sampling plans.
            


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ACC702 Management Accounting

Question 1 You are to create a mathematical model of the motion of the fire pot as it is swung in a circle by the goblin. Your model must describe the trajectory of the fire pot in the camera frame of reference: that is, provide and equation for the displacement of the fire pot, as a function of tim ...

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RSA cryptography

Question 1 (6 marks) A description of RSA cryptography is given at the end of the lecture notes for Section B1.(Numbers). The topic is there described as non-assessed. I should have added that the description relates to examinations and regular weekly assignments, but that it is a suitable topic for ...

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FM 5001 Preparation for Financial Mathematics

Question: 1. How many five-digit zip codes can be made where all digits are unique? Assume all digits 0 – 9 are allowed. What is the probability of the zip code being “53820”?2. Suppose that 35 people are divided in a random manner into two teams in such a way that one team contai ...

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Management

A small business makes cookies and sells them all at the farmers market. The fixed monthly cost for use of a health department approved kitchen and rental space at the farmers market is $790..... Write a linear cost function representing the cost C(x) to produce x dozen cookies per month. Write a li ...

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