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Chapter 5: Problem 15

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 \(=50 \mathrm{mg} / \mathrm{dL}\). Some investigators believe that only cholesterol levels over \(250 \mathrm{mg} / \mathrm{dL}\) indicate a high-enough risk for heart disease to warrant treatment. What proportion of the population does this group represent?

Short Answer

Expert verified
26.76% of the population has cholesterol levels above 250 mg/dL.

Step by step solution

01

Understand the Problem

We need to find the proportion of people with cholesterol levels over 250 mg/dL. This requires finding the probability that a value from a normal distribution with mean 219 mg/dL and standard deviation 50 mg/dL exceeds 250 mg/dL.
02

Standardize the Cholesterol Level

To find this probability, we first standardize the cholesterol level using the z-score formula: \[ z = \frac{X - \mu}{\sigma} \]where \(X = 250\), \(\mu = 219\), and \(\sigma = 50\). Substituting these values in, we get:\[ z = \frac{250 - 219}{50} = 0.62 \]
03

Use the Standard Normal Distribution

Once we have the z-score, we can use the standard normal distribution table, or a calculator, to find the probability that \(Z > 0.62\). This gives us the proportion of the population with cholesterol levels over 250 mg/dL.
04

Find the Complementary Probability

The probability \(P(Z > 0.62)\) is the complementary probability of \(P(Z \leq 0.62)\). From standard normal distribution tables or using a calculator, we find that \(P(Z \leq 0.62)\) is approximately 0.7324. Thus, the probability that \(Z > 0.62\) is:\[ P(Z > 0.62) = 1 - P(Z \leq 0.62) = 1 - 0.7324 = 0.2676 \]
05

Answer the Original Question

About 26.76% of the population has cholesterol levels over 250 mg/dL, indicating high risk for heart disease according to the investigators' threshold.

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Key Concepts

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

Normal Distribution
A normal distribution is a fundamental concept in statistics. It is often referred to as a "bell curve" due to its distinct shape. Normal distributions are symmetrical and centered around a mean value, with data tapering off equally on both sides. For many natural phenomena, such as cholesterol levels in humans, data is approximately normally distributed.
  • The mean represents the average value of all data points.
  • The standard deviation indicates how much variation or "spread" there is from the mean.
In the context of cholesterol, understanding the normal distribution helps us predict the proportion of individuals within certain cholesterol ranges. This kind of distribution allows for the use of statistical techniques to make meaningful inferences about population data.
Z-score
The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is expressed in terms of standard deviations from the mean. Calculating a z-score is crucial when analyzing normally distributed data as it transforms the values into a standard form.
  • The z-score formula is given by: \[ z = \frac{X - \mu}{\sigma} \]
  • \(X\) is the value of interest, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Z-scores allow comparisons between different data sets or within a single set to understand how unusual or typical a certain observation is. In this exercise, we used it to determine how much higher a cholesterol level of 250 mg/dL is compared to the average.
Probability
Probability is the measure of the likelihood that an event will occur. In statistics, it quantifies certainty or uncertainty associated with outcomes in given scenarios. By using the normal distribution and z-score, we can calculate probabilities.
  • A probability of 1 indicates a certain event.
  • A probability of 0 means the event will not occur.
In this exercise, once we calculated the z-score for a cholesterol level of 250 mg/dL, we determined the probability of an individual having levels above this by referencing the standard normal distribution. This probability represents the proportion of the population potentially at higher risk for heart disease.
Cholesterol Levels
Cholesterol levels are key biomarkers used to gauge cardiovascular health. Cholesterol is necessary for building cells and producing hormones, but excessive levels can be detrimental. Levels are typically measured in milligrams per deciliter (mg/dL).
  • Normal range tends to be less than 200 mg/dL.
  • Levels above 240 mg/dL are typically considered high.
In this exercise, the focus is on those with levels over 250 mg/dL, a threshold considered risky by some researchers. Since cholesterol naturally varies in a population, understanding its distribution and calculating those at risk helps in prevention and treatment planning.
Heart Disease Risk
Heart disease remains a leading cause of death globally, and managing risk factors is crucial. One significant factor is cholesterol level, with higher levels associated with increased plaque build-up in arteries. When cholesterol levels exceed thresholds like 250 mg/dL, the risk of heart disease can increase substantially.
  • Monitoring and controlling cholesterol is part of preventive strategies.
  • Dietary changes and medication may be recommended for high-risk individuals.
This exercise underscores the importance of understanding statistical distributions to identify and manage elevated risk in populations, which ultimately aids in proactive healthcare measures.

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Most popular questions from this chapter

Suppose for the sake of simplicity that the incidence of macular degeneration is \(1 \%\) per year among people \(65+\) years of age in the therapeutic range \((\geq 10 \mathrm{mg} / \mathrm{dL})\) and \(2 \%\) per year among people \(65+\) years of age with lower levels of lutein \((<10 \mathrm{mg} / \mathrm{dL})\). What is the expected relative risk of macular degeneration for lutein-treated participants versus placebo-treated participants in the proposed study?

Because serum cholesterol is related to age and sex, some investigators prefer to express it in terms of \(z\) -scores. If \(X=\) raw serum cholesterol, then $$Z=\frac{X-\mu}{\sigma}$$, where \(\mu\) is the mean and \(\sigma\) is the standard deviation of serum cholesterol for a given age-gender group. Suppose \(Z\) is regarded as a standard normal random variable. What is \(\operatorname{Pr}(Z>0.5) ?\)

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