Question

A population has a mean of 84 and a standard deviation of 12. A sample of 36 observations will be taken. The probability that the sample mean will be between 80.54 and 88.9 is

Answer 1

Answer:

The probability that the sample mean will be between 80.54 and 88.9 is 0.951

Step-by-step explanation:

* Lets revise some definition to solve the problem

- The mean of the distribution of sample means is called M

- The standard deviation of the distribution of sample means is

 called σM

- σM = σ/√n , where σ is the standard deviation and n is the sample size

- z-score = (M - μ)/σM, where μ is the mean of the population  

* Lets solve the problem

∵ The sample size n = 36

∵ The sample mean M is between 80.54 and 88.9

∵ The mean of population μ = 84

∵ The standard deviation σ = 12

- Lets find σM to find z-score  

∵ σM = σ/√n

∴ σM = 12/√36 = 12/6 = 2

- Lets find z-score

∵ z-score = (M - μ)/σM

∴ z-score = (80.54 - 84)/2 = -3.46/2 = -1.73

∴ z-score = (88.9 - 84)/2 = 4.9/2 = 2.45

- Use the normal distribution table to find the probability

∵ P(-1.73 < z < 2.45) = P(2.45) - P(-1.73)

∴ P(-1.73 < z < 2.45) = 0.99286 - 0.04182 = 0.95104

∴ P(-1.73 < z < 2.45) = 0.951

* The probability that the sample mean will be between 80.54 and 88.9

  is 0.951