Use a normal approximation to find the probability of the indicated number of vo

Question

Use a normal approximation to find the probability of the indicated number of voters. In this case, assume that 133 eligible voters aged 18-24 are randomly selected. Suppose a previous study showed that among eligible voters aged 18-24, 22% of them voted. Probability that fewer than 33 voted The probability that fewer than 33 of 133 eligible voters voted is? Round to four decimal places. I try the problem on my own first and i did. 133(0.22)=29.26 then I did the square root of 133*0.22(1-0.22)=4.777321425 round it to 4.777 next I did (33-29.26)/(4.777)= 0.7829. I got this answer and it was still wrong.

Explanation / Answer

let X be the random variable denoting the number of voters who have voted among the 133 eligible voters selected.

A previous study showed that among eligible voters aged 18-24, 22% of them voted.

so X~Bin(133,0.22)

so E[X]=133*0.22 and V[X]=133*0.22*(1-0.22)

now since n=133 is very high, the distribution of X can be approximated by a normal distribution with mean 133*0.22 and variance 133*0.22*(1-0.22)

so X~N(29.26,22.8228)

so P[less than 33 voted]=P[X<33]=P[(X-29.26)/sqrt[22.8228]<(33-29.26)/sqrt[22.8228]]=P[Z<=0.782865]=0.7831468 [answer]

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