Decide whether you can use the normal distribution to approximate the binomial d

Question

Decide whether you can use the normal distribution to approximate the binomial distribution If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. you cannot, explain why and use the binomial distribution to find the indicated probabilities. Five percent of workers in a city use public transportation to get to work You randomly select 279 workers and ask them if they use public transportation lo get to parts (a) through (d). (b) Find the probability that at least 8 workers will say yes. What is the indicated probability? (Round to four decimal places as needed) sketch the graph of the normal distribution with the indicated probability shaded. The normal distribution cannot be 13.95 used to approximate the binomial distribution.

Explanation / Answer

given that ,

n=279

asking question yes or no ( p=q=0.5 )

x follow binomial distribution (n,p)

we find the probability that at least 8 workers will say that yes,

P(x>=8)=1-P(x<8)

=1- [p(x=0)+p(x=1)+p(x=2)+p(x=3)+p(x=4)+p(x=5)+p(x=6)+p(x=7)]

=1-[(279C0*(0.5)^0*(0.5)^279-0 )+27(279C1*(0.5)^1*(0.5)^279-1)+(279C2*(0.5)^2*(0.5)^279-2)+(279C3*(0.5)^3*(0.5)^279-3)+(279C4*(0.5)^4*(0.5)^279-4)+(279C5*(0.5)^5*(0.5)^279-5)+(279C6*(0.5)^6*(0.5)^279-6)+(279C7*(0.5)^7*(0.5)^279-7)]

= 1-[4.11*10^-67]

=1

apply central limit theoram,

z=(x-mean(x))/var(x) follo standard normal distribution ,

mean(x)=n*p

=279*0.5

=139.5 so that option (d) is correct, mean is 139.5 (not 13.95).

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