SOLUTION: Historically, 93% of the deliveries of an overnight mail service arrive before 10:30 the following morning. If a random sample of 500 deliveries is selected, what is the probabilit

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Question 1195851: Historically, 93% of the deliveries of an overnight mail service arrive before 10:30 the following morning. If a random sample of 500 deliveries is selected, what is the probability the sample will have more than 95% of the deliveries arriving before 10:30 the following morning?
Found 2 solutions by Boreal, ikleyn:
Answer by Boreal(15235)   (Show Source): You can put this solution on YOUR website!
This is calculating a z-value which is here z=(phat-p)/sqrt (p*(1-p)/n)
=0.02/sqrt(0.93*0.07/500)
=0.02/0.0114
=1.75
Probability z > 1.75 is 0.0398
-
Can check with a normal approximation
mean is 0.93*500=465
variance is that *0.07=32.55
sd is sqrt of that=5.71
95% of 500 is 475
so z=(475.5-465)/5.71, using the continuity correction factor
=1.84 and the probability of z> 1.84 is 0.0329.
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
Historically, 93% of the deliveries of an overnight mail service arrive before 10:30
the following morning. If a random sample of 500 deliveries is selected,
what is the probability the sample will have more than 95% of the deliveries arriving
before 10:30 the following morning?
~~~~~~~~~~~~~


            In order for the mathematical and the probabilistic meaning of this problem
            would become more clear,  let me re-formulate it in this way. 


                There is a binomial distribution with the individual probability of success of 0.93.

                There is a random sample of 500 trials.

                What is the probability of more than 0.95*500 = 475 successful trials ?



In this formulation, the probability is a standard binomial distribution problem

    P = P(n=500; k>=476; p=0.93).


To facilitate calculations, I use online calculator at this site  https://stattrek.com/online-calculator/binomial.aspx

     It provides nice instructions  and  a convenient input and output for all relevant options/cases.


          The resulting number is P = 0.02796  (rounded).    ANSWER

Solved.



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