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 _____.
A) .0347
B) .7200
C) .9511
D) None of the answers is correct.

Question

Quizplus · Accepted Answer

To solve this problem, we need to use the formula for the standard error of the mean: SE = σ/√n where σ is the population standard deviation, n is the sample size, and SE is the standard error of the mean. In this case, σ = 12 and n = 36, so: SE = 12/√36 = 2 Next, we need to standardize the sample mean using the z-score formula: z = (x - μ) / SE where x is the sample mean, μ is the population mean, and SE is the standard error of the mean. We want to find the probability that the sample mean falls between 80.54 and 88.9, so we need to standardize those values as well: z1 = (80.54 - 84) / 2 = -1.73 z2 = (88.9 - 84) / 2 = 2.45 Now we can look up these z-scores in the standard normal distribution table (or use a calculator) to find the probabilities: P(z < -1.73) = 0.0422 P(z > 2.45) = 0.0070 Note that we need to use the complement of P(z > 2.45) to find the probability that the sample mean is less than 88.9, so: P(z < 2.45) = 0.9929 P( -1.73 < z < 2.45) = 0.9929 - 0.0422 = 0.9507 Therefore, the probability that the sample mean will be between 80.54 and 88.9 is 0.9507, which is approximately 0.9511. The answer is C.

A Population Has a Mean of 84 and a Standard