Question

The annual precipitation amounts in a certain mountain range are normally distributed with a mean of 91 inches, and a standard deviation of 14 inches. what is the probability that the mean annual precipitation during 49 randomly picked years will be less than 93.8 inches?

Answer 1

To find the probability that the mean annual precipitation during 49 randomly picked years will be less than 93.8 inches, we use the properties of the normal distribution.

Step-by-step explanation:

To find the probability that the mean annual precipitation during 49 randomly picked years will be less than 93.8 inches, we can use the properties of the normal distribution.

First, we need to calculate the standard error of the mean, which is the standard deviation divided by the square root of the sample size. In this case, the standard error is 14/sqrt(49) = 2 inches.

Next, we can convert the desired mean of 93.8 inches to a z-score using the formula (x - mean)/standard error. Plugging in the values, we get a z-score of (93.8 - 91)/2 = 1.4.

Using a standard normal distribution table or a calculator, we can find the probability that a z-score is less than 1.4, which is approximately 0.919.

Answer 2

The sample std. dev. will be (14 inches) / sqrt(49), or (14 inches) / 7, or 2 inches.

Find the z score for 93.8 inches:
93.8 inches - 91.0 inches 2.8 inches
z = ------------------------------------- = ----------------- = 1.4
2 inches 2 inches

Now find the area under the standard normal curve to the left of z = +1.4.

My calculator returns the following:

normalcdf(-100,1.4) = 0.919. This is the probability that the mean annual precipitation during those 49 years will be less than 93.8 inches.