Question

Use a normal approximation to find the probability of the indicated number of voters. In this​ case, assume that 146 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 37 voted

Answer 1

Solution:

Given:  146 eligible voters aged​ 18-24 are randomly selected.

Thus n = 146

Among eligible voters aged​ 18-24, 22% of them voted

thus p = probability of voted = 0.22

We have to use  a normal approximation to find the probability that fewer than 37 voted.

P( X < 37) = ............?

Since X = Number of people voted follows a Binomial distribution with parameters n = 146 and p = 0.22

mean = n*p = 146 * 0.22 = 32.12

and

Standard Deviation is:

Now use continuity correction to find the probability

that is add or subtract 0.5 from x value.

here we have X < 37, so we need to subtract 0.5 from 37 , in order to exclude from the range of x values.

Thus we get : 37-0.5=36.5

thus we have to find: P( X < 36.5 )

Now find z score for x = 36.5

Thus we get:

P( X< 36.5) =P( Z < 0.88)

Look in z table for z = 0.8 and 0.08 and find corresponding area.

P( Z< 0.88 ) =0.8106

Thus

P( X< 36.5) =P( Z < 0.88)

P( X< 36.5) = 0.8106

that is:

P( X < 37) = 0.8106