Question

A normally distributed population has a mean of 500 and a standard deviation of 60. Determine the probability that a random sample of size 25 selected from the population will have a sample mean greater than or equal to 515.

Answer 1

The value is
`P(\= X \ge 515 ) = 0.8944`

Explanation:

From the question we are told that

The mean is
`\mu = 500`

The standard deviation is
`\sigma = 60`

The sample size is n = 25

Generally the standard error of mean is mathematically represented as

`\sigma _(\= x ) = (\sigma )/(√(n) )`

=>
`\sigma _(\= x ) = (60)/(√(25) )`

=>
`\sigma _(\= x ) = 12`

Generally the probability that a random sample of size 25 selected from the population will have a sample mean greater than or equal to 515 is mathematically represented as

`P(\= X \ge 515 ) = 1 - P(\= X < 515)`

Here
`P(\= X < 515) = P((\= x - \mu )/(\sigma_(\= x )) < (515 - 500)/(12) )`

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

`P(\= X < 515) = P(Z < 1.25 )`

From the z table the probability of (Z < 1.25 )

`P(Z < 1.25 ) = 0.10565`

So

`P(\= X < 515) = 0.10565`

So

`P(\= X \ge 515 ) = 1 - 0.10565`

=>
`P(\= X \ge 515 ) = 0.8944`