# 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.

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

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 following probabilities (in order good, average,

defective):

0, 10, 0

1, 8, 1

2, 6, 2

3, 4, 3

4, 2, 4

5, 0, 5

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?

You can use a difference equation.

Let E = expected number of throws to a double 6

You MUST throw at least once and there is 35/36 probability of

returning to the start point:

E = 1 + (35/36)E

E(1 - 35/36) = 1

E(1/36) = 1

E = 36

So the expected number of throws is 36.

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?

Proportion defective = 1/5 = 0.2 while claim is 0.1

p = 0.1

pq/n = 0.1 x 0.9/15 = 0.006

sqrt(pq/n) = 0.07746

We test

0.2 - 0.1

z = --------- = 1.291

0.07746

and we compare this with 1.645 (single tailed test), and see that it

is not significant. So do not reject null hypothesis.

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

correct.

p=q = 1/2

npq = (1/2)(1/2)(200) = 50

sqrt(npq) = 7.071

110 - 100 10

z = --------- = ----- = 1.4142

7.071 7.071

A(z) = 0.9214

so area from mean to this value is 0.4214. By symmetry area other side

of the mean is also 0.4214. Total area corresponding to range 90 - 110

is 0.8428, so we can accept null hypothesis with probability 0.8428.

b) Find the probability of rejecting the hypothesis when it is

actually correct.

This is simply 1 - the previous answer.

1 - 0.8428 = 0.1572

BEST OF LUCK!

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