Question

If 51% of a population will vote for Candidate A in an election, what is the probability that in a random sample of 75 voters, fewer than 50% of the sample will vote for Candidate A

Answer 1

The value is
`P(X < 0.50 ) = 0.43133`

Explanation

From the question we are told that

The population proportion is p = 0.51

The sample size is n = 75

Generally given that the sample size is large enough (i.e n > 30) the mean of this sampling distribution is mathematically represented as

`\mu_x = p = 0.5 1`

Generally the standard deviation of this sample distribution is mathematically represented as

`\sigma = \sqrt{(p(1- p))/( n) }`

=>
`\sigma = \sqrt{(0.51 (1- 0.51 ))/(75) }`

=>
`\sigma = 0.058`

Generally the probability that in a random sample of 75 voters, fewer than 50% of the sample will vote for Candidate A is mathematically represented as

`P(X < 0.50 ) = P( ( X - \mu_x )/(\sigma ) < ( 0.50 - 0.51 )/(0.0578 ) )`

`(X -\mu)/(\sigma )  =  Z (The  \ standardized \  value\  of  \ X )`

=>
`P(X < 0.50 ) = P( Z < -0.1730 )`

From the z table the area under the normal curve to the left corresponding to -0.1720 is

`P( Z < -0.1730 ) = 0.43133`

So

`P(X < 0.50 ) = 0.43133`