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Answer:
0.8944 = 89.44% probability that the mean annual precipitation during 25 randomly picked years will be less than 112 inches.
Step-by-step explanation:
To solve this question, we use the normal probability distribution and the central limit theorem.
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean and standard deviation
, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Mean of 109.0 inches, and a standard deviation of 12 inches.
This means that
Sample of 25.
This means that
What is the probability that the mean annual precipitation during 25 randomly picked years will be less than 112 inches?
This is the p-value of Z when X = 112. So
By the Central Limit Theorem
has a p-value of 0.8944.
0.8944 = 89.44% probability that the mean annual precipitation during 25 randomly picked years will be less than 112 inches.