Ordering Probabilities

Date: 08/16/97 at 12:34:49
From: Amy Hunter
Subject: Statistics

In a restaurant, the proportion of people who order coffee with their 
dinner is .9.  A simple random sample of 144 patrons of the restaurant 
is taken. 

What is the expected value, standard deviation, and shape of the 
sampling distribution of p?

What is the probability that the proportion of people who will order 
coffee with their meal is between .85 and .875?

What is the probability that the proportion of people who will order 
coffee with their meal is at least .945?

I don't understand the proportions and probabilities.  Thanks.

Date: 08/16/97 at 17:14:34
From: Doctor Anthony
Subject: Re: Statistics

>What is the expected value, standard deviation, and shape of the sampling distribution of p?

This is a binomial probability with mean = np = 144 x 0.9 = 129.6
                                variance = npq = 144 x .9 x .1 = 12.96
                                     s.d = sqrt(12.96) = 3.6

If dealing with proportions of successes,  mean = p = 0.9
                            variance = pq/n = .9 x .1/144 = .000625
                                 s.d = sqrt(pq/n) = .025
                        

>What is the probability that the proportion of people who will order coffee with their meal is between .85 and .875?

        .85 - .9
  z1 = ----------  = - 2.0     A(z1) = .9772 
          .025

         .875 - .9
  z2 =  -----------  = - 1.0    A(z2) = .8413
           .025

The area we require is that between z1 and z2 = .9772 - .8413 = 0.1359

So the probability that the proportion is between .85 and .875  
is 0.1359.

>What is the probability that the proportion of people who will order coffee with their meal is at least .945?

        .945 - .9
  z  =  ---------   = 1.8      A(z) = 0.9641
          .025  

The probability that the proportion is .945 or greater is 
1 - .9641 = .0359

-Doctor Anthony,  The Math Forum
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