Question

A population of values has a normal distribution with μ=235μ=235 and σ=69.8σ=69.8. You intend to draw a random sample of size n=145n=145.

Find the probability that a single randomly selected value is between 232.1 and 247.2.
P(232.1 < X < 247.2) =

Find the probability that a sample of size n=145n=145 is randomly selected with a mean between 232.1 and 247.2.
P(232.1 < ¯¯¯XX¯ < 247.2) =

Enter your answers as numbers accurate to 4 decimal places.

Answer 1

Solution :

Given that ,

mean = = 235

standard deviation = = 69.8

a) P(232.1 < x < 247.2) = P[(232.1 - 235)/ 69.8) < (x - ) /  < (247.2 - 235) / 69.8 ) ]

= P(-0.04 < z < 0.17)

= P(z < 0.17) - P(z < -0.04)

Using z table,

= 0.5675 - 0.4840

= 0.0835

b) n = 145

= = 235

= / n = 69.8 / 145 = 5.80

P(232.1 < < 247.2)  

= P[(232.1 - 235) / 5.80 < ( - ) / < (247.2 - 235) / 5.80)]

= P( -0.50 < Z < 2.10)

= P(Z < 2.10) - P(Z < -0.50)

Using z table,  

= 0.9821 - 0.3085

= 0.6736