Question

The SAT scores for US high school students are normally distributed with a mean of 1500 and a standard deviation of 100.

1. Calculate the probability that a randomly selected student has a SAT score greater than 1650.

2. Calculate the probability that a randomly selected student has a SAT score between 1400 and 1650, inclusive.

3. If we have random sample of 100 students, find the probability that the mean scores between 1485 and 1510, inclusive.

Answer 1

a)Solution :

Given ,

mean = = 1500

standard deviation = = 100

P(x >1650 ) = 1 - P(x< 1650)

= 1 - P[ X - / / (1650-1500) / 100]

= 1 - P(z <1.5 )

Using z table

= 1 - 0.9332

= 0.0668

probability=0.0668

b)

P(1400< x <1650 ) = P[(1400-1500) /100 < (x - ) / < (1650-1500) / 100)]

= P(-1 < Z < 1.5)

= P(Z < 1.5) - P(Z < -1)

Using z table   

= 0.9332 - 0.1587

probability= 0.7745

c)

Solution :

Given that ,

mean =   = 1500

standard deviation = = 100   

n = 100

= 1500

=  / n= 100 / 100=10

P(1485<     <1510 ) = P[(1485-1500) /10 < ( - ) /   < (1510-1500) /10 )]

= P(-1.5 < Z <1 )

= P(Z < 1) - P(Z < -1.5)

Using z table

=0.8413 -0.0668

=0.7745

probability= 0.7745