Question

1. Suppose the lengths of all fish in an area are normally distributed, with mean μ = 29 cm and standard deviation σ = 4 cm. What is the probability that a fish caught in an area will be between the following lengths? (Round your answers to four decimal places.)
(a) 29 and 37 cm long

(b) 24 and 29 cm long

2.  

The Scholastic Aptitude Test (SAT) scores in mathematics at a certain high school are normally distributed, with a mean of 500 and a standard deviation of 100. What is the probability that an individual chosen at random has the following scores? (Round your answers to four decimal places.)
(a) greater than 600

(b) less than 400

(c) between 550 and 700

Answer 1

Solution :

Given that ,

1) mean = = 29

standard deviation = = 4

a . P(29 < x < 37) = P[(29-29)/4 ) < (x - ) /  < (37- 29) /4 ) ]

= P(0 < z <2 )

= P(z < 2) - P(z < 0)

= 0.9772-0.5 =0.4772

probability = 0.4772

b .

P(24 < x < 29) = P[(24-29)/4 ) < (x - ) /  < (29- 29) /4 ) ]

= P(-1.25 < z <0 )

= P(z < 0) - P(z < -1.25)

= 0.5 - 0.4772 =0.0228

probability = 0.0228

2 )

mean = = 500

standard deviation = = 100

a.P(x >600 ) = 1 - P(x < 600 )

= 1 - P[(x - ) / < (600- 500) /100 ]

= 1 - P(z < 1)

=1 -0.8413=0.1587

probability = 0.1587

b.

P(x < 400) = P[(x - ) / < (400-500) /100 ]

= P(z < -1) = 0.1587

probability =0.1587

c)

P(550 < x < 700) = P[550 - 500/100 ) < (x - ) /  < (700-500) /100 ) ]

= P(0.5 < z <02)

= P(z < 2) - P(z < 0.5)

= 0.9772 - 0.6915 =0.2857

probability = 0.2857