The serum cholesterol levels for men in one age group are normally distributed with a mean of 178.1 and a standard deviation 40.5. All units are in mg/10 mL. Find the 40th percentile
Question
The serum cholesterol levels for men in one age group are normally distributed with a mean of 178.1 and a standard deviation 40.5. All units are in mg/10 mL. Find the 40th percentile
Solution
To find the 40th percentile of a normally distributed set of data, we need to use the concept of z-scores. A z-score tells us how many standard deviations an element is from the mean.
The z-score for the 40th percentile (which is the same as 0.40 in decimal form) can be found in a standard normal distribution table or can be calculated using a calculator or statistical software. The z-score for the 0.40 probability/percentile is approximately -0.25.
Once we have the z-score, we can use the formula for transforming a z-score into an original score (X):
X = μ + Zσ
where:
- X is the score we're looking for,
- μ is the mean,
- Z is the z-score, and
- σ is the standard deviation.
Substituting the given values into the formula, we get:
X = 178.1 + (-0.25)(40.5)
Calculate the product and addition to find X. This will give you the serum cholesterol level corresponding to the 40th percentile.
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