Given :- A previous study showed that among eligible voters aged 18-24, 22% of them voted.
154 eligible voters aged 18-24 are randomly selected.
Therefore,
Probability that voters aged 18-24 will vote = P = 0.22
Sample size of voters = n = 154.
Let Binomial variable X be "The number of voters aged 18-24 who voted" out of 154.
probability that exactly 39 voted , so we are asked to find P(X = 39).
Rule :- If n*P and n*q both are greater than 5 Then varible X is approximated by Normal distribution with mean µ = n*P and standard deviation σ = √n*p*q.
Let us check if above neccessary condition is satisfied or not,
n*P = 154 * 0.22 = 33.88. n*q = 154 * 0.78 = 120.12
Both 33.88 and 120.12 are greater than 5 so we can approximate variable X by "Normal distribution".
Also,
mean µ = n*P = 154 * 0.22 = 33.88
Standard deviation σ = √n*p*q = σ = √154 * 0.22 * 0.78 = 5.1406.
Continuity correction Factor :- Binomial variable X = 39 is a discrete number, But we are approximating it to a continuous Normal variable.
So to convert a discrete number 39 into continuous format, we use continuity correction factor. (kindly find the image attached for more examples).
By using continuity correction factor we have,
P(X = 39) = P(38.5 ≤ X ≤ 39.5)
= P(0.9 < Z < 1.09) (by using Z score formula)
= P(Z < 1.09) - P(Z < 0.9)
= 0.8621 - 0.8159 ( by using z score table )
= 0.0462
