Expectation and Hypothesis Testing

Date: 08/24/99 at 11:27:24
From: Dave Wright
Subject: Statistics: Multinomial, Neg Binom, Hyp Test

I have a few 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.
Dave Wright

Date: 08/24/99 at 18:35:39
From: Doctor Anthony
Subject: Re: Statistics: Multinomial, Neg Binom, Hyp 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 

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/