Calculating Probability with Normal Approximation

[lilyungn]

Hi all I'm currently taking Statistics 1 and I'm stuck on a homework problem that I've been trying to figure out for a while...hopefully one of you guys can enlighten me on how to do it..heres the question:

Transportation officials tell us that 80% of drivers wear seat belts while driving. What is the probability of observing 518 or fewer drivers wearing seat belts in a sample of 700 drivers?

Hint: Use normal distribution to approximate the binomial distribution

Any help will be greatly appreciated. Thanks

 

[CRGreathouse]

The normal approximation of the binomial distribution has mean n * p and variance n * p * (1 - p). In your example n = 700 and p = 0.8.

I calculate the exact probability as

27215033424606239489951580591428932253089201021148168958765691731392094513257400149512461174994101101893879349950817724535023320833098115143452605318902844562357900521627169600482502350655645510044817056094897193166242140544070444191441905644096259918364809861980951251235754139967622866482769375805893409967413796426281723416461645206238828172038546224013601381995986790697562819660604903531492876045502982090073701576428830644538188684724158694295977900811634082990023633233789421341/380218313259031964703014481167020621852974241274703806488349211513170849855494440711440299410524372640522125900659637734955652488757724700708924916770505595637569416206346918582471539076959056068435223651008041554876432039792771230305103084765331596853800084507271724280140484161410403670670196458543283896996507022028449859411384981297711080778428012558385332383706122652531298594333598103700679951981973698960512606011735240635218234741615222116624395187756135783274658024311065673828125

You can use this to check yourself, if you'd like.

 

[lilyungn]

I don't understand how you did it, can you show the work? thanks

 

[Hurkyl]

I don't understand how you did it, can you show the work? thanks

No, we cannot; that is cheating, strictly against site policy, and not very effective in helping you learn anyways.

What we will do is help you solve the problem, but we can't do that if you only post the question, and nothing about your own thoughts, work, and understanding about it.

 

[Mark44]

The normal approximation of the binomial distribution has mean n * p and variance n * p * (1 - p). In your example n = 700 and p = 0.8.

I calculate the exact probability as

27215033424606239489951580591428932253089201021148168958765691731392094513257400149512461174994101101893879349950817724535023320833098115143452605318902844562357900521627169600482502350655645510044817056094897193166242140544070444191441905644096259918364809861980951251235754139967622866482769375805893409967413796426281723416461645206238828172038546224013601381995986790697562819660604903531492876045502982090073701576428830644538188684724158694295977900811634082990023633233789421341/380218313259031964703014481167020621852974241274703806488349211513170849855494440711440299410524372640522125900659637734955652488757724700708924916770505595637569416206346918582471539076959056068435223651008041554876432039792771230305103084765331596853800084507271724280140484161410403670670196458543283896996507022028449859411384981297711080778428012558385332383706122652531298594333598103700679951981973698960512606011735240635218234741615222116624395187756135783274658024311065673828125

You can use this to check yourself, if you'd like.

Seems a little large for a probability, being that they should be in the interval [0, 1].