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Answer:
0.0268 = 2.68% probability that the sample average is greater than $229,500
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
When the distribution is normal, we use 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.
Single house:
The mean appraised value of a house in this subdivision for all houses is $228,000, with a standard deviation of $8,500, which means that
Sample:
Sample of 120, so, by the central limit theorem,
What is the probability that the sample average is greater than $229,500?
This is 1 subtracted by the pvalue of Z when X = 229.5. So
By the Central Limit Theorem
has a pvalue of 0.9732
1 - 0.9732 = 0.0268
0.0268 = 2.68% probability that the sample average is greater than $229,500
f(x) is the same as y, so we can say y = f(x)
Writing f(x) < 0 means we want to find when y < 0.
Visually, we are looking at the graph when the curve is below the horizontal x axis.
This is the portion in red that I have marked in the diagram (see attached image below). I apologize for the numbers being blurry.
The left red portion is from negative infinity to -3. In terms of a compound inequality we write which in interval notation is
. The curved parenthesis tells the reader to exclude both endpoints.
The right red portion is from x = -1.1 to x = 0.9, excluding both endpoints. So we say which becomes the interval notation
. This is not ordered pair notation even though it looks identical to it.
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<h3>Answer in interval notation:The "U" means "set union" which glues together the two separate intervals. Basically it's saying "x is either in the interval (-infinity, -3) or it is in the interval (-1.1, 0.9)"
