Question

The proportion of eligible voters in the next election who will vote for the incumbent is assumed to be 53.6%. What is the probability that in a random sample of 470 voters, less than 48.1% say they will vote for the incumbent

Answer 1

The required probability = 0.0084

Explanation:

GIven that:

Sample proportion = 53.6% = 0.536

Sample size = 470

Then; the mean
`\mu` = n × p

mean
`\mu` = 470 × 0.536

mean
`\mu` = 251.92

Standard deviation =
`√(n* p(1-p))`

Standard deviation =
`√(470 * 0.536(1-0.536))`

Standard deviation =
`√(470 * 0.536(0.464))`

Standard deviation =
`√(116.89088)`

Standard deviation = 10.81

The sample mean
`\overline x = n * (48.1)/(100)`

The sample mean
`\overline x = 470 * (48.1)/(100)`

The sample mean
`\overline x` = 226.07

Thus;

`P( \overline x < 226.07) = P(Z < (\overline x - \mu )/(\sigma))`

`P( \overline x < 226.07) = P(Z < (226.07 - 251.92 )/(10.81))`

`P( \overline x < 226.07) = P(Z < -2.39)`

`P( \overline x < 226.07)` = 0.0084

Therefore, the required probability = 0.0084