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

Answer:

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

Step-by-step 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

       $\sigma _{\= x } = \frac{\sigma }{\sqrt{n} }$

=>    $\sigma _{\= x } = \frac{60}{\sqrt{25} }$

=>    $\sigma _{\= x } = 12$

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  $P(\= X < 515) = P(\frac{\= x - \mu }{\sigma_{\= x }} < \frac{515 - 500}{12} )$

$\frac{X -\mu}{\sigma }  =  Z (The  \ standardized \  value\  of  \ X )$

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

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

    $P(Z < 1.25 ) = 0.10565$

So

    $P(\= X < 515) = 0.10565$

So

    $P(\= X \ge 515 ) = 1 - 0.10565$

=>  $P(\= X \ge 515 ) = 0.8944$