A set of data has a normal distribution with a mean of 180 and a standard deviation of 20. What percent of the data is in the interval 140 - 220?

allucinemsj 2022-08-13 Answered
A set of data has a normal distribution with a mean of 180 and a standard deviation of 20. What percent of the data is in the interval 140 - 220?
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Answers (1)

Macie Melton
Answered 2022-08-14 Author has 19 answers
We are given the information that this distribution is normal with a mean μ of 180 and a standard deviation σ of 20. This describes a distribution N ( 180 , 20 2 ).
To answer the question, we will convert this problem into a standard normal distribution N s ( 0 , 1 2 ) question by determining the z-scores for the interval 140-220. This can be done either by "eyeballing it" (since σ is 20, and each endpoint of the interval is a multiple of σ away from the mean μ), or we can use the z-score formula:
z = x μ σ
Thus:
z 140 = 140 180 20 = 40 20 = 2 z 140 = 220 180 20 = 40 20 = 2
This tells us the interval we're being asked about is analogous to determining what percent of the standard normal distribution N s lies between z-scores of -2 and 2.
In statistics there is a handy "rule of thumb" sometimes called the Empirical Rule which says the approximately 95% of the data in a normal distribution lies in the interval [ 2 σ , 2 σ ], which is exactly what we're being asked. (The actual answer is more like 95.45%.)
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