The incomes in a certain large population of college teachers have a normal distribution with mean $75,000 and standard deviation $10,000. Four teachers are selected at random from this population to serve on a salary review committee. The probability that their average salary is less than $65,000 is closest to .

The probability that their average salary is less than $65,000 is closest to 2.28%.

Step-by-step explanation:

To solve this question, we have 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 , a large sample size can be approximated to a normal distribution with mean and standard deviation, which is also called standard error

In this problem, we have that:

The probability that their average salary is less than $65,000 is closest to

This is the pvalue of Z when X = 65000. So

By the Central Limit Theorem

has a pvalue of 0.0228.

The probability that their average salary is less than $65,000 is closest to 2.28%.

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(credits to leowren)

step-by-step explanation: