Question

1. Scores on a certain nationwide college entrance examination follow a normal distribution with a mean of 500 and a standard deviation of 100. Find the probability that a student will score (a) over 650. Answer: (b) less than 459. Answer: (C) between 325 and 675. Answer: (d) If a school only admits students who score over 680, what proportion of the students pool would be eligible for admission? Answer: (e) What limit (score) would you set that makes the top 20% of the students eligible? (Choose closest number in table. Do not interpolate). Answer: (1) If 16 examinations are selected at random and their mean is calculated, i. what is the expected average score? Answer: ii. what is the standard deviation for the average score? Answer:

Answer 1

Solution :

Given that ,

mean = $\mu$ = 500

standard deviation = $\sigma$ = 100

P(x > 650 ) = 1 - P( x< 650 )

= 1- P[(x - $\mu$ ) / $\sigma$ < ( 650 - 500 ) / 100 ]

= 1- P(z < 1.5 )

Using z table,

= 1 - 0.9332

= 0.0668

Answer = Probability = 0.0668

( b )

P(x < 459 )

= P[(x - $\mu$ ) / $\sigma$ < ( 459 - 500) / 100]

= P(z < -0.41 )

Using z table,

= 0.3409

Answer = Probability = 0.3409

( c )

P( 325 < x < 675 )

= P[( 325 - 500 ) / 100 ) < (x - $\mu$ ) /$\sigma$  < ( 675 - 500 ) / 100 ) ]

= P( -1.75 < z < 1.75)

= P(z < 1.75 ) - P(z < -1.75)

Using z table,

= 0.9599 - 0.0401

= 0.9198

Answer = Probability = 0.9198

( d )

P(x > 680) = 1 - P( x < 680 )

= 1- P[(x - $\mu$ ) / $\sigma$ < ( 680 - 500) / 100 ]

= 1- P(z < 1.8)

Using z table,

= 1 - 0.9641

= 0.0359

Answer = proportion = 0.0359

( e )

The z - distribution of the 20% is

P(Z > z) = 20%

= 1 - P(Z < z ) = 0.20

= P(Z < ) = 1 - 0.20

= P(Z < z ) = 0.80

= P(Z < 0.842 )

z = 0.842

Using z-score formula,

x = z * $\sigma$ + $\mu$

x = 0.842 * 100 + 500

x = 584.2

Answer = x = 584.2

( f )

n = 16

$\mu$$\bar x$ = 500

$\sigma$$\bar x$ = $\sigma$ / $\sqrt$ n = 100 / $\sqrt$ 16  = 25

The average score $\mu$ $\bar x$ = 500

The standard deviation for the average score $\sigma$ $\bar x$ = 25