Question

SAT verbal scores are normally distributed with a mean of 430 and a standard deviation of 120 (based on the data from the College Board ATP). If a sample of 15 students is selected randomly, find the probability that the sample mean is above 500. Does the central limit theorem apply for this problem?

Answer 1

Answer:

The value is  $P(X > 500 ) = 0.011942$

Step-by-step explanation:

From the question we are told that

       The mean is  $\mu = 430$

       The standard deviation is  $\sigma = 120$

       The sample size is  n  =  15

   

Generally  the probability that the sample mean is above 500

Gnerally the standard error of mean is mathematically represented as

             $s = \frac{\sigma }{ \sqrt{n} }$

=>          $s = \frac{ 120 }{ \sqrt{15} }$

=>           $s = 30.984$

Generally  the probability that the sample mean is above 500

     $P(X > 500 ) = P( \frac{X - \mu }{\sigma_x} > \frac{500 - 430 }{ 30.984 } )$

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

       $P(X > 500 ) = P( Z > 2.259 )$

=>     $P(X > 500 ) = P( Z > 2.259 )$

From the z table  the area under the normal curve to the right  corresponding to  2.259  is

          $P( Z > 2.259 ) = 0.011942$

=>       $P(X > 500 ) = 0.011942$