Question

A population is normally distributed with μ = 200 and σ = 20.
a. Find the probability that a value randomly selected from this population will have a value greater than 250.

P(x > 250) = ______ . (Round to four decimal places as needed.)

b. Find the probability that a value randomly selected from this population will have a value less than 185.

P(x < 185) = ______ . (Round to four decimal places as needed.)

c. Find the probability that a value randomly selected from this population will have a value between 185 and 250.

P(185 < x < 250) = ______ . (Round to four decimal places as needed.)

Answer 1

Solution :

Given that ,

mean = $\mu$ = 200

standard deviation = $\sigma$ = 20

a) P(x > 250) = 1 - P(x < 250)

= 1 - P[(x - $\mu$ ) / $\sigma$ < (250 - 200 ) /20 )

= 1 - P(z < 2.5)

= 1 - 0.9938

0.0062

P(x > 250) = 0.0062

b) P(x < 185) = P[(x - $\mu$ ) / $\sigma$ < (185 - 200) /20]

= P(z < -0.75)

= 0.2266

P(x < 185) = 0.2266

c) P(185 < x < 250) = P[(185 - 200)/ 20) < (x - $\mu$ ) /$\sigma$  < (250 - 200) / 20) ]

= P(-0.75 < z < 2.5)

= P(z < 2.5) - P(z < -0.75)

= 0.9938 - 0.2266

0.7672

P(185 < x < 250) = 0.7672