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Question 1

An importer of laptops has three suppliers: A1, A2 and A3. 20% of the laptops are supplied by A1, 70% are supplied by A2 and 10% are supplied by A3. From historical data it has been established that the percentage of defective laptops from suppliers A1, A2 and A3 is 12%, 3% and 2% respectively.

Calculate the probability of selecting a defective laptop

n1: 20

n2: 70

n3: 10

K1: 12

K2: 3

K3: 2

= .6854

Given that a customer returns a defective laptop, calculate the probability that the laptop was supplied by i) A1, ii) A2 iii) A3

A1 = .256

ii) A2 = .0582 iii) A3= .761

Question 2

It is known that one out of every five tax returns will contain errors and will be classed as faulty.

Calculate the probability that in a sample of five tax returns

none are faulty

0.32768

ii) at least one is faulty

.00854

Calculate the probability that at most two tax returns are faulty

0.94208

An inspector randomly selects a sample of 20 tax returns. Calculate the probabilities that in the sample of 20

i) seven are faulty = .684

ii) at most two are faulty = .0215

State the assumptions underlying the probability distribution you used to answer parts a)-c) above.

The Binomial Distribution Formula

where

n = the number of trials (or the number being sampled)

x = the number of successes desired

p = the probability of getting a success in one trial

q = 1 - p = the probability of getting a failure in one trial.

Assumptions of the binomial distribution

The experiment involves n identical trials.

Each trial has only two possible outcomes denoted as success or failure.

Each trial is independent of the previous trials.

The terms p and q remain constant throughout the experiment, where p is the probability of getting a success on any one trial and q = (1 - p) is the probability of getting a failure on any one trial.

Suppose now that the total population of tax returns in a country is one million and you could assume a sample of 250,000 of them to calculate the probability that at least 10% of them are faulty. Would the answer you provided in part d) still hold? Why (not)?

Use the binomial test when you have dichotomous data - that is, when each individual in the sample is classified in one of two categories (e.g. category A and category B) and you want to know if the proportion of individuals falling in each category differs from chance or from some pre-specified probabilities of falling into those categories.

Assumptions: The normal approximation for the Binomial test assumes that the proportion of the time that individuals are expected to fall into category A (symbolized by "p") multiplied by the total number of individuals in category A and B combined (symbolized by "n") is greater than 10 (i.e. pn>10) and that the proportion of the time that individuals are expected to fall into category B (symbolized by "q") multiplied by the total number of individuals ...