Question

1. Suppose the scores on a chemistry test were normally distributed with a mean of 78 and a standard deviation of 10. If a student who completed the test is chosen at random,

a. Find the probability that the student earned fewer than 75 points.

b. Find the probability that the student earned at least 70 points.

c. Find the probability that the student earned between 80 and 90 points.

d. Find the probability that the student earned either less than 80 points or more than 90 points.

Answer 1

Solution :

Given that,

mean = = 78

standard deviation = = 10

a ) P( x < 75 )

P ( x - / ) < ( 75 - 78 / 10)

P ( z < - 3 / 10 )

P ( z < - 0.30 )

= 0.3821

Probability = 0.3821

b ) P (x 70 )

= 1 - P (x 70 )

= 1 - P ( x -  / ) ( 70 - 78 / 10)

= 1 - P ( z - 8 / 10 )

= 1 - P ( z -0.80)

Using z table

= 1 - 0.2119

= 0.7881

Probability = 0.7881

c ) P ( 80 < x < 90 )

P ( 80 - 78 / 10) < ( x -  / ) < ( 90 - 78 / 10)

P ( 2 / 10 < z < 12 / 10 )

P (0.20 < z < 1.20 )

P ( z < 1.20 ) - P ( z < 0.20)

Using z table

= 0.8849 - 0.5793

= 0.3056

Probability = 0.3056

d ) P( x < 80 )

P ( x - / ) < ( 80 - 78 / 10)

P ( z < 2 / 10 )

P ( z < 0.20 )

= 0.5793

P (x > 90 )

= 1 - P (x < 90 )

= 1 - P ( x -  / ) < ( 90 - 78 / 10)

= 1 - P ( z < 12 / 10 )

= 1 - P ( z < 1.20)

Using z table

= 1 - 0.8849

= 0.1151

Probability = 0.5793 + 0.1151 = 0.6944