Answer:
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:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
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 ![\mu = 40000, \sigma = 2000](https://tex.z-dn.net/?f=%5Cmu%20%3D%2040000%2C%20%5Csigma%20%3D%202000)
Jean selected three neighborhoods and determined the average income level.
This means that ![n = 3, s = \frac{2000}{\sqrt{3}} = 1154.7](https://tex.z-dn.net/?f=n%20%3D%203%2C%20s%20%3D%20%5Cfrac%7B2000%7D%7B%5Csqrt%7B3%7D%7D%20%3D%201154.7)
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
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
By the Central Limit Theorem
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{38000 - 40000}{1154.7}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B38000%20-%2040000%7D%7B1154.7%7D)
![Z = -1.73](https://tex.z-dn.net/?f=Z%20%3D%20-1.73)
has a pvalue of 0.0418
0.0418 = 4.18% probability that the average income level in the neighborhoods was less than $38,000.