Question

1a) Assume we're working with normally distributed SAT scores, with the following distribution: Mean = 500 and Std. Dev = 100. A student is randomly selected from the SAT population. What's the probability of that student's score being between 350 and 600?

1b) Assume we're working with normally distributed SAT scores, with the following distribution: Mean = 500 and Std. Dev = 100. A college decides to admit students with SAT scores greater than or equal to 450. Assuming the applicant population contains 1500 students, how many would be admitted?

1c) Assume we're working with normally distributed SAT scores, with the following distribution: Mean = 500 and Std. Dev = 100. A college decides to admit only the top 10% of SAT students. What would its cutoff SAT score be?

Answer 1

Solution :

Given that ,

1 a)mean = = 500

standard deviation = = 100

P(350 < x < 600 ) = P[(300-500)/100 ) < (x - ) /  < (600-500) /100 ) ]

= P(-2 < z <1 )

= P(z <1 ) - P(z <-2 )

= 0.8289 - 0.0202 =0.8088

Probability = 0.8088

1 b)

P(x 450 ) = 1 - P(x   450)

= 1 - P[(x - ) / (450-500) /100 ]

= 1 -  P(z -0.50)  

= 1- 0.2912 = 0.7088

Probability = 0.7088 *1500 = 1063.2

Answer = 1063 admitted.

1 c) top 10 %

P(Z > z ) = 0.10

1- P(z < z) =0.10

P(z < z) = 1-0.10 = 0.90

z = 1.282

Using z-score formula,

x = z * +

x = 1.282*100+500

x = 628.2

cutoff SAT score is = 628.2