Question

A population of values has a normal distribution with μ=124.2 and σ=75.5. You intend to draw a random sample of size n=35

Find the probability that a single randomly selected value is between 121.6 and 145.9.
P(121.6 < X < 145.9) = ____________

Find the probability that a sample of size n=35 is randomly selected with a mean between 121.6 and 145.9.
P(121.6 < M < 145.9) = __________

Enter your answers as numbers accurate to 4 decimal places.

Answer 1

Solution :

Given that ,

mean = = 124.2

standard deviation = = 75.5

a)

P( 121.6 < x < 145.9) = P((121.6 - 124.2)/ 75.5) < (x - ) /  < (145.9 - 124.2) / 75.5) )

= P(-0.03 < z < 0.29)

= P(z < 0.29) - P(z < -0.03)

= 0.6141 - 0.4880 Using standard normal table,  

Probability = 0.1261

b)

n = 35

= 124.2

= / n = 75.5 / 35 = 12.7618

P(121.6 < M < 145.9) = P((121.6 - 124.2) /12.7618 <(M - ) / < (145.9 - 124.2) / 12.7618))

= P(-0.20 < Z < 1.70)

= P(Z < 1.70) - P(Z < -0.20) Using standard normal table,  

= 0.9554 - 0.4207

=0.5347

Probability = 0.5347