Question

S.M.A.R.T. test scores are standardized to produce a normal distribution with a mean of 230 and a standard deviation of 35. Find the proportion of the population in each of the following S.M.A.R.T. categories.

1. Genius: Score of greater than 330.

2. Superior Intelligence: Score between 280 and 330.

3. Average intelligence: Score between 201 and 260.

Please show all work in equations.

Answer 1

Solution:

Given that,

= 230

= 35

1) p ( x > 330 )

= 1 - p (x < 330 )

= 1 - p ( x -  / ) < ( 330 - 230 / 35)

= 1 - p ( z < 100 / 35 )

= 1 - p ( z < 2.86)

Using z table

= 1 - 0.9979

= 0.0021

Probability = 0.0021

2 ) p (280 < x  < 330 )

= p( 280 - 230 / 35 ) ( x -  / ) < ( 330 - 230 /35)

= p ( 50 / 35 < z < 100 / 35 )

= p ( 1.43 < z < 2.86 )

= p (z < 2.86 ) - p ( z < 1.43 )

Using z table

= 0.9979 - 0.9236

= 0.0743

Probability = 0.0743

3 ) p (201 < x  < 260 )

= p( 201 - 230 / 35 ) ( x -  / ) < ( 260 - 230 /35)

= p ( - 29 / 35 < z < 30 / 35 )

= p ( - 0.83 < z < 0.86 )

= p (z < 0.86 ) - p ( z < - 0.83)

Using z table

= 0.8051 - 0.2033

= 0.6018

Probability = 0.6018