Question

A population of values has a normal distribution with μ = 240.5 and σ = 70.2 . You intend to draw a random sample of size n = 131 .

Find the probability that a single randomly selected value is between 246 and 249.7. P(246 < X < 249.7)

Find the probability that a sample of size n = 131 is randomly selected with a mean between 246 and 249.7. P(246 < M < 249.7)

Answer 1

Solution :

Given that ,

mean = = 240.5

standard deviation = = 70.2

1)

P(246 < x < 249.7) = P((246 - 240.5)/ 70.2) < (x - ) /  < (249.7 - 240.5) / 70.2) )

= P(0.08 < z < 0.13)

= P(z < 0.13) - P(z < 0.08)    Using z table,

= 0.5517 - 0.5319

= 0.0198

The probability that a single randomly selected value is between 246 and 249.7 is 0.0198.

2)

n = 131

=M =   = 240.5

= / n = 70.2/ 131 = 6.1334

P(246 < M < 249.7) = P((246 - 240.5)/ 6.1334) < (M - ) /  < (249.7 - 240.5) / 6.1334)

= P(0.90 < Z <1.50)

= P(Z < 1.50) - P(Z < 0.90) Using z table,

= 0.9332 - 0.8159

= 0.1173

The probability that a sample of size n = 131 is randomly selected with a mean between 246 and 249.7 is 0.1173