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