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Multinomial, Negative Binomials and Hypothesis Testing

I have several questions I'm stuck on:

1. A production line produces good articles with probability .7,
average ones with probability of .2, and defective ones with
probability .1. Ten articles are selected.

a) What is the probability of 8 good ones and 1 defective?

b) What is the probability that there is an equal number of good
and defective articles?

Answers in back of text: a) 0.1036 b) 0.006165

I attempted to solve this using Multinomial Distribution.

2. A person decides to throw a pair of dice until he gets 2 sixes.
What is the expected number of throws until he stops?

Answer in back of text: 36

I attempted to solve this using Negative Binomial Distribution.

3. A transistor manufacturer claims its product has 10% defectives.
A sample of 15 transistors is examined and 3 are found to be
defective. Would you reject their claim with alpha <= 0.05?

Answer in back of text: No

4. An experiment is set to test the hypothesis that a given coin is
unbiased. The decision rule is the following: Accept the hypothesis
if the number of heads in a sample of 200 tosses is between 90 and
110 inclusive, otherwise reject the hypothesis.

a) Find the probability of accepting the hypothesis when it is
correct.

b) Find the probability of rejecting the hypothesis when it is
actually correct.

Answer in back of text: a) 0.8612 b) 0.1388

Thanks in advance.

Solution Preview

Please refer to response file attached (also see below). I hope this helps and take care.

RESPONSE:

Statistics: Multinomial, Negative Binomial, Hypothesis Test

1. a) What is the probability of 8 good ones and 1 defective?

10!
P(8 good, 1 av, 1 defective) = -------- x 0.7^8 x 0.2 x 0.1
8! 1! 1!

= 90 x 0.7^8 x 0.2 x 0.1

= 0.103766

b) What is the probability that there is an equal number of good
and defective articles?

Work out the ...

Solution Summary

This solution provides a step--by-step solution to four statistics questions concerning multinomial, negative binomials and
hypothesis testing.

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