Question

The life of a new gadget you invented to water a plant is normally distributed with a mean of 3 days with a standard deviation of 10 hours.   You took a sample of 16 gadgets.

A. What is the probability that the sample mean will be more than 3.2 days?

B. What is the probability that the sample mean would be between 2.8 and 3.2?

C. What is the probability that the sample mean would be either less than 2.75 or greater than 2.75?

Answer 1

Let X denotes the life of the new gadget invented to water a plant. Then X follows normal distribution with mean 3 days(72 hours) and standard deviation 10 hours. If X follows normal distribution with mean and standard deviation then the sample mean follows normal distribution with mean mean and standard deviation according to the central limit theorem. Here n = 16. So the sample mean of 16 gadgets follow normal distribution with mean 72 hours and standard deviation hours.

The probability can be evaluated using the standard normal transformation

A.

The probability that the sample mean of 16 gadgets will be more than 3.2 days(76.8 hours) is given as

The probability that the sample mean life of the 16 gadgets will be more than 3.2 days is 0.02743.

B.

The probability that the sample mean of 16 gadgets will be between 2.8 days (67.2 hours) and 3.2 days(76.8 hours) is given as

= 2(0.97257) - 1

= 0.94514

The probability that the sample mean of 16 gadgets will be between 2.8 days (67.2 hours) and 3.2 days(76.8 hours) is 0.94514

C.

The probability that the sample mean of 16 gadgets would be either less than 2.75(66 hours) or greater than 2.75(66 hours) will be obviously one as the probability will cover the entire region.

The probability that the sample mean would be either less than 2.75 or greater than 2.75 is 1.