Jean works for the government and was conducting a survey to determine the income levels of a number of different neighborhoods in a metropolitan area. Based on national data, Jean knows that the mean income level in the country is $40,000, with a standard deviation of $2,000. Jean selected three neighborhoods and determined the average income level. What is the probability that the average income level in the neighborhoods was less than $38,000

0.0418 = 4.18% probability that the average income level in the neighborhoods was less than $38,000.

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 normal distributions can be 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 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.

Jean knows that the mean income level in the country is $40,000, with a standard deviation of $2,000.

This means that

Jean selected three neighborhoods and determined the average income level.

This means that

What is the probability that the average income level in the neighborhoods was less than $38,000

This is the pvalue of Z when X = 38000. So

By the Central Limit Theorem

has a pvalue of 0.0418

0.0418 = 4.18% probability that the average income level in the neighborhoods was less than $38,000.

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