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Chapter 5: Problem 14

Serum cholesterol is an important risk factor for coronary disease. We can show that serum cholesterol is approximately normally distributed, with mean \(=219 \mathrm{mg} / \mathrm{dL}\) and standard deviation \(=50 \mathrm{mg} / \mathrm{dL}\). If the clinically desirable range for cholesterol is \(<200 \mathrm{mg} / \mathrm{dL},\) what proportion of people have clinically desirable levels of cholesterol?

Short Answer

Expert verified
35.20% of people have clinically desirable cholesterol levels.

Step by step solution

01

Understand the Normal Distribution

The question states that serum cholesterol levels are approximately normally distributed. In a normal distribution, the probability of a value falling within a range can be found using the z-score formula and standard normal distribution tables or a calculator.
02

Calculate the Z-Score

The z-score formula is: \[ z = \frac{X - \mu}{\sigma} \]Where:- \( X \) is the desired cholesterol level, in this case, 200 mg/dL.- \( \mu \) is the mean, which is 219 mg/dL.- \( \sigma \) is the standard deviation, which is 50 mg/dL.Substituting the values, we get:\[ z = \frac{200 - 219}{50} = \frac{-19}{50} = -0.38 \]
03

Use the Z-Score to Find Proportion

The z-score of -0.38 represents the number of standard deviations below the mean 200 mg/dL is. Look up this z-score in the standard normal distribution table, or use a calculator, which gives a proportion of approximately 0.3520.
04

Interpret the Result

The probability associated with a z-score of -0.38 corresponds to the area under the normal curve to the left of 200 mg/dL. This tells us that approximately 35.20% of people have cholesterol levels less than 200 mg/dL, which is in the clinically desirable range.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Serum Cholesterol
Serum cholesterol is a crucial aspect of our health as it serves as a risk marker for coronary artery disease. This type of cholesterol flows in your bloodstream and can significantly impact cardiovascular health. While cholesterol itself is necessary for building healthy cells, having high levels can increase the risk of heart diseases.
Unlike dietary cholesterol that we consume directly from food, serum cholesterol is primarily produced by the liver. It is essential to monitor its levels to reduce health risks.
  • Cholesterol is transported in the blood by lipoproteins. These can be classified mainly as low-density (LDL) and high-density lipoproteins (HDL).
  • LDL is often referred to as "bad cholesterol," as high levels can lead to plaque buildup in arteries and result in heart disease.
  • HDL is known as "good cholesterol" because it helps transport cholesterol away from the arteries and back to the liver, where it is either removed from the body or reused.
This understanding is fundamental when assessing risks and devising strategies for maintaining cholesterol within healthy ranges.
Cholesterol Levels and Their Significance
Understanding cholesterol levels is vital for determining the risk of developing heart disease. Cholesterol levels are measured in milligrams per deciliter of blood (mg/dL). What is fascinating is that cholesterol levels in the population follow a normal distribution, which allows us to make certain predictions.
  • If cholesterol levels are below 200 mg/dL, they are considered clinically desirable or healthy, minimizing the risk of heart disease.
  • Levels ranging from 200 to 239 mg/dL are considered borderline high, indicating a moderate risk.
  • Levels of 240 mg/dL and above are deemed high, substantially increasing the risk of coronary diseases.
Since cholesterol levels follow a normal distribution, most people will have levels close to the mean. However, understanding the distribution allows us to calculate the percentage of individuals maintaining levels within the desired range, thereby guiding public health decisions.
Mastering Z-Score Calculation
The z-score is a statistical measure that describes a value's relation to the mean of a group of values. Here's how it plays a role in assessing cholesterol levels. When we want to understand what proportion of people have cholesterol levels in a certain range, such as clinically desirable levels below 200 mg/dL, we use the z-score.To calculate the z-score, use the formula:\[ z = \frac{X - \mu}{\sigma} \]- **\(X\)** is the value of interest (200 mg/dL in this context).
- **\(\mu\)** is the mean (219 mg/dL in this case).
- **\(\sigma\)** is the standard deviation (50 mg/dL).In this example, by substituting the provided values, the z-score calculation is:\[ z = \frac{200 - 219}{50} = -0.38 \]The z-score of -0.38 indicates the number of standard deviations the value (200 mg/dL) is below the mean. Using a standard normal distribution table or calculator, a z-score of -0.38 corresponds to a proportion of approximately 0.3520.This translates to roughly 35.20% of people having cholesterol levels within the clinically desirable range of under 200 mg/dL. Such calculations help in understanding the population's health status and assist health professionals in making informed decisions.

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Most popular questions from this chapter

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