The annual precipitation amounts in a certain mountain range are normally distributed with a mean of 104 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 106.8 inches

91.92% probability that the mean annual precipitation during 49 randomly picked years will be less than 106.8 inches

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central limit theorem:

The Central Limit Theorem estabilishes that, for a random variable X, with mean and standard deviation , the sample means with size n of at least 30 can be approximated to a normal distribution with mean and standard deviation

In this problem, we have that:

What is the probability that the mean annual precipitation during 49 randomly picked years will be less than 106.8 inches

This is the pvalue of Z when X = 106.8. So

By the Central Limit Theorem

has a pvalue of 0.9192

0.9192 = 91.92% probability that the mean annual precipitation during 49 randomly picked years will be less than 106.8 inches

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step-by-step explanation: