Question

Given a normal population whose mean is 645 and whose standard deviation is 36, find each of the following:

A. The probability that a random sample of 7 has a mean between 651 and 655.

Probability =

B. The probability that a random sample of 14 has a mean between 651 and 655.

Probability =

C. The probability that a random sample of 27 has a mean between 651 and 655.

Probability =

Answer 1

Solution :

Given that,

A)

mean = = 645

standard deviation = = 36

n = 7

= = 645

= / n = 36 / 7 = 13.6067

P( 651 < < 655) = P((651 - 645) / 13.6067 <( - ) / < (655 - 645) / 13.6067))

= P(0.29 < Z < 0.49)

= P(Z <0.49) - P(Z < 0.29) Using z table,

= 0.6879 - 0.6141

= 0.0738

Probability = 0.0738

B)

mean = = 645

standard deviation = = 36

n = 14

= = 645

= / n = 36 / 14 = 9.6214

P( 651 < < 655) = P((651 - 645) / 9.6214 <( - ) / < (655 - 645) / 9.6214))

= P(0.62 < Z < 1.04)

= P(Z <1.04) - P(Z < 0.62) Using z table,

= 0.8508 - 0.7324

= 0.1184

Probability = 0.1184

C)

mean = = 645

standard deviation = = 36

n = 27

= = 645

= / n = 36 / 27 = 6.9282

P( 651 < < 655) = P((651 - 645) / 6.9282 <( - ) / < (655 - 645) / 6.9282))

= P(0.87 < Z < 1.44)

= P(Z <1.44) - P(Z < 0.87) Using z table,

= 0.9251- 0.8078

= 0.1173

Probability = 0.1173