Question

Use a normal approximation to find the probability of the indicated number of voters. in this​ case, assume that 154 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 exactly 39 voted

Answer 1

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