Question

Given a population where the probability of success is p= 0.40​, calculate the probabilities below if a sample of is taken. Calculate the probability the proportion of successes in the sample will be . What is the probability the proportion of successes in the sample will be ​?

Answer 1

Complete Question

Given a population where the probability of success is p= 0.40​ calculate the probabilities below if a sample of 300 is taken.

A. Calculate the probability the proportion of successes in the sample will be less than 0.42 (round 4 decimals)

B. What is the probability that the proportion of successes in the sample will be greater than 0.44 (round 4 decimals)

Answer:

A

`P(X < 0.42) = 0.76028`

B

`P(X > 0.44) = 0.078622`

Explanation:

From the question we are told that

The probability of success is p = 0.40

The sample size is n = 300

Generally given that the sample size is large enough n > 30 then the mean for this sampling distribution is

`\mu_(x) = p = 0.40`

Generally the standard deviation is mathematically represented as

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

=>
`\sigma = \sqrt{ (0.40 (1 - 0.40 ))/( 300) }`

=>
`\sigma = 0.02828`

Considering question A

Generally the probability the proportion of successes in the sample will be less than 0.42 is mathematically represented as

`P(X < 0.42) = P((X - \mu )/(\sigma ) < (0.42 - 0.40 )/( 0.02828) )`

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

=>
`P(X < 0.42) = P(Z < 0.7072 )`

From the z table

The area under the normal curve to the left corresponding to 0.7072 is

`P(Z < 0.7072 ) = 0.76028`

=>
`P(X < 0.42) = 0.76028`

Considering question B

Generally the probability the proportion of successes in the sample will be less than 0.44 is mathematically represented as

`P(X > 0.44) = P((X - \mu )/(\sigma ) > (0.44 - 0.40 )/( 0.02828) )`

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

=>
`P(X > 0.44) = P(Z > 1.4144 )`

From the z table

The area under the normal curve to the left corresponding to 1.4144 is

`P(Z > 1.4144 ) = 0.078622`

=>
`P(X > 0.44) = 0.078622`