Solve the problem.
-The annual precipitation amounts in a certain mountain range are normally distributed with a mean of 109 inches, and a standard deviation of 10 inches. What is the probability that the mean annual precipitation during
25 randomly picked years will be less than 111.8 inches?
A) 0.0808
B) 0.5808
C) 0.9192
D) 0.4192

Question

Quizplus · Accepted Answer

We use the central limit theorem since we are sampling means from a normal distribution. The mean of the sampling distribution will be the same as the population mean (109 inches) and the standard deviation of the sampling distribution will be the population standard deviation divided by the square root of the sample size (10 inches/sqrt(25) = 2 inches).

We want to find the probability of the sample mean being less than 111.8 inches, so we standardize this value using the formula:

z = (x̄ - μ) / (σ / sqrt(n))

z = (111.8 - 109) / (10 / sqrt(25))

z = 1.8/2 = 0.9

Then, we look up the probability of a z-score less than 0.9 in a standard normal distribution table or calculator, which is approximately 0.8192.

However, we need to use the one-tailed probability since the question is asking for the probability of the mean being less than 111.8 inches.

P(z < 0.9) = 0.5 + 0.3192 = 0.8192

So the probability that the mean annual precipitation during 25 randomly picked years will be less than 111.8 inches is 0.9192 or approximately 0.92 when rounded to two decimal places. Therefore, the best choice is C.

Solve the Problem. -The Annual Precipitation Amounts in a Certain Mountain Range Are