Question

Student scores on Professor Combs' Stats final exam are normally distributed with a mean of 72 and a standard deviation of 7.5

Find the probability of the following:

**(use 4 decimal places)**

a.) The probability that one student chosen at random scores above an 77.  

b.) The probability that 20 students chosen at random have a mean score above an 77.  

c.) The probability that one student chosen at random scores between a 67 and an 77.  

d.) The probability that 20 students chosen at random have a mean score between a 67 and an 77.

Answer 1

Solution :

Given that ,

mean = =  72

standard deviation = = 7.5

(A)n = 1

= 72

= / n = 7.5/ 1 = 7.5

P( >77 ) = 1 - P( < 77)

= 1 - P[( - ) / < (77- 72) /7.5 ]

= 1 - P(z <0.67 )

Using z table

= 1 - 0.7486

= 0.2514

probability= 0.2514

(B)

n = 20

= 72

= / n = 7.5/ 20 = 1.6771

P( >77 ) = 1 - P( < 77)

= 1 - P[( - ) / < (77- 72) /1.6771 ]

= 1 - P(z <2.99)

Using z table

= 1 - 0.9986

= 0.0014

probability= 0.0014

(C)

n = 1

= 72

= / n = 7.5/ 1 = 7.5

P(67<     <77) = P[(67-72) / 7.5< ( - ) /   < (77-72) / 7.5)]

= P( -0.67< Z < 0.67)

= P(Z <0.67 ) - P(Z <- 0.67)

Using z table

=0.7486-0.2514

=0.4972

probability= 0.4972

(D)

n = 20

= 72

= / n = 7.5/ 20 = 1.6771

P(67<     <77) = P[(67-72) / 1.6771< ( - ) /   < (77-72) / 1.6771)]

= P( -2.98< Z < 2.98)

= P(Z <2.98 ) - P(Z <- 2.98)

Using z table

=0.9986-0.0014

=0.9972

probability= 0.9972