Question

The mean of a population is 74 and the standard deviation is 16. The shape of the population is unknown. Determine the probability of each of the following occurring from this population.

a. A random sample of size 32 yielding a sample mean of 78 or more
b. A random sample of size 130 yielding a sample mean of between 71 and 76
c. A random sample of size 219 yielding a sample mean of less than 74.7

(Round all the values of z to 2 decimal places and final answers to 4 decimal places.)

Answer 1

The values provided in the question are as below

Population mean = = 74

Standard deviation = = 16

We determine the probability of each of the following occurring from this population.

a. A random sample of size 32 yielding a sample mean of 78 or more

Here, sample size = n = 32

We have to find the probability that a random sample of size 32 yielding a sample mean of 78 or more

----------(1)

We convert above into z using following formula

-----------(2)

Using equation (2) in equation (1) we get

We round above z up to 2 decimal places

We find above probabiliy using z table of standard normal curve areas

  

The probability that a random sample of size 32 yielding a sample mean of 78 or more is 0.0793

b. A random sample of size 130 yielding a sample mean of between 71 and 76

Here, sample size = n = 130

We have to find the probability that a random sample of size 130 yielding a sample mean of between 71 and 76

----------(3)

We convert above into z using following formula

-----------(4)

Using equation (4) in equation (3) we get

We round above z up to 2 decimal places

We find above probabiliy using z table of standard normal curve areas

  

The probability that a random sample of size 130 yielding a sample mean of between 71 and 76 is 0.0602

c. A random sample of size 219 yielding a sample mean of less than 74.7

Here, sample size = n = 219

We have to find the probability that a random sample of size 219 yielding a sample mean of less than 74.7

-----------(5)

We convert above into z using following formula

-------------(6)

Using equation (6) in equation (5) we get

We round above z up to 2 decimal places

We find above probabiliy using z table of standard normal curve areas

The probability that a random sample of size 219 yielding a sample mean of less than 74.7 is 0.7422

Summary :-

a. The probability that a random sample of size 32 yielding a sample mean of 78 or more is 0.0793

b. The probability that a random sample of size 130 yielding a sample mean of between 71 and 76 is 0.0602

c. The probability that a random sample of size 219 yielding a sample mean of less than 74.7 is 0.7422