Problem 74 - Applied Statistics and Probability for Engineers

Question

Cholesterol is a fatty substance that is an important part of the outer lining (membrane) of cells in the body of animals. Its normal range for an adult is \(120-240 \mathrm{mg} / \mathrm{dl}\). The Food and Nutrition Institute of the Philippines found that the total cholesterol level for Filipino adults has a mean of \(159.2 \mathrm{mg} / \mathrm{dl}\) and \(84.1 \%\) of adults have a cholesterol level less than \(200 \mathrm{mg} / \mathrm{dl}\) (http://www.fnri.dost.gov.ph/). Suppose that the total cholesterol level is normally distributed. (a) Determine the standard deviation of this distribution. (b) What are the quartiles (the \(25 \%\) and \(75 \%\) percentiles) of this distribution? (c) What is the value of the cholesterol level that exceeds \(90 \%\) of the population? (d) An adult is at moderate risk if cholesterol level is more than one but less than two standard deviations above the mean. What percentage of the population is at moderate risk according to this criterion? (e) An adult whose cholesterol level is more than two standard deviations above the mean is thought to be at high risk. What percentage of the population is at high risk? (f) An adult whose cholesterol level is less than one standard deviations below the mean is thought to be at low risk. What percentage of the population is at low risk?

Step-by-Step Solution

Answer

(a) Standard deviation is approximately 40.8 mg/dl. (b) Quartiles are 131.7 mg/dl and 186.7 mg/dl. (c) 90th percentile is 211.3 mg/dl. (d) 34.13% are at moderate risk. (e) 2.28% are at high risk. (f) 15.87% are at low risk.

Step 1: Understanding Normal Distribution Properties

A normal distribution is characterized by its mean (\(\mu\)) and standard deviation (\(\sigma\)). The area under the curve corresponds to probabilities and percentages of data points within certain intervals.

Step 2: Determine the Standard Deviation (σ)

Given that the 84.1% of adults have cholesterol levels below 200 mg/dl, we can use the z-score formula: \(z = \frac{X - \mu}{\sigma}\). Using standard normal distribution tables or a calculator, we find the z-score for 0.841, which is approximately 0.995. Set \(159.2 + 0.995\sigma = 200\), and solve for \(\sigma\): \(\sigma = \frac{200 - 159.2}{0.995} \approx 40.8\).

Step 3: Calculate the Quartiles

For the 25th quartile (percentile 25%), the z-score is approximately -0.675. For the 75th quartile (percentile 75%), the z-score is about 0.675. Calculate them: Quartile1 \(= 159.2 + (-0.675) \times 40.8 \approx 131.7\) mg/dl; Quartile3 \(= 159.2 + 0.675 \times 40.8 \approx 186.7\) mg/dl.

Step 4: Determine the 90th Percentile

Locate the z-score that corresponds to the 90th percentile, which is about 1.28. The cholesterol level \(X\) is given by \(X = 159.2 + 1.28\times 40.8 = 211.344\) mg/dl.

Step 5: Calculate Moderate Risk Percentage

Moderate risk is defined as having a cholesterol level between 1 and 2 standard deviations above the mean. The z-scores are 1 and 2. Calculate probabilities using normal distribution, which gives an area between 1 and 2 standard deviations (34.13%).

Step 6: Determine High Risk Percentage

Adults at high risk have cholesterol levels more than 2 standard deviations above the mean. From the normal distribution, the probability is about 2.28% for z > 2.

Step 7: Calculate Low Risk Percentage

Low risk is defined as a cholesterol level less than 1 standard deviation below the mean. Calculate using z-score -1: the percentage is about 15.87%.

Key Concepts

Cholesterol Levels

Cholesterol is a crucial component of the cell membranes in animal bodies and plays a significant role in human health. Understanding cholesterol levels is important because they can be an indicator of cardiovascular health. For adults, a typical cholesterol range is between 120 and 240 mg/dl. This wide range covers what is considered normal and safe for many people, but individual health conditions might call for more specific targets.

In our exercise, the average cholesterol level for Filipino adults is 159.2 mg/dl. This mean value tells us that this is the center point of the data if it follows a normal distribution. However, not everyone's cholesterol level will be exactly at this mean. Instead, there's variation among individuals, and that's where the normal distribution helps to explain how these levels spread across a population. By understanding this distribution, we can identify how typical or atypical certain cholesterol levels are within a given group.

Standard Deviation

The standard deviation (σ) is a measure of how spread out the numbers are in a data set relative to the mean (μ). In the context of cholesterol levels, it provides insight into how much variation there is among individuals' levels around the average of 159.2 mg/dl. A higher standard deviation means a wider range of cholesterol levels among people.

For our question, we found the standard deviation to be approximately 40.8 mg/dl. This was calculated by using the z-score, which is a statistical measure that tells us how many standard deviations an element is from the mean. With the information that 84.1% of adults have levels below 200 mg/dl, we used a z-score table to find the z-score corresponding to 84.1%, which is roughly 0.995. We then used this z-score in the formula \( z = \frac{X - \mu}{\sigma} \)to solve for the standard deviation.

This calculation lets us know how diversified the cholesterol levels are in the sample population, offering a fuller picture of how much an individual might deviate from the average.

Percentiles

Percentiles are a way to understand the relative standing of a value within a data set. When we talk about the 25th percentile, it means that 25% of the data points fall below this value, while the 75th percentile means 75% are below it. These are also known as quartiles, with Q1 at 25% and Q3 at 75%. Finding the percentiles helps us understand the distribution of cholesterol levels better.

In the exercise, we calculated these quartiles to be approximately 131.7 mg/dl for Q1 and 186.7 mg/dl for Q3. To find these, we used z-score values of -0.675 for the 25th percentile and 0.675 for the 75th percentile. These scores were plugged into the formula for a normal distribution to find the corresponding cholesterol level.

This means if someone’s cholesterol is 131.7 mg/dl, they have lower cholesterol than 25% of the population. At 186.7 mg/dl, they have lower cholesterol than 75%. Understanding where values fall in percentiles helps in assessing risk levels and making appropriate health recommendations.

Z-Scores

Z-scores are a statistical metric that gives insight into an individual data point's position within a normal distribution. Essentially, it tells us how many standard deviations away from the mean a data point is. Positive z-scores indicate values above the mean, while negative scores are below.

In the context of cholesterol levels, using z-scores helps to determine where a particular cholesterol level stands when compared to the average cholesterol level in the population. For instance, if a cholesterol level corresponds to a z-score above 1.28, it is higher than 90% of the population's cholesterol levels, which we found to correspond to around 211.344 mg/dl.

Z-scores are particularly useful for defining risk categories. By calculating where the levels fall in terms of standard deviations, we can classify them as high, moderate, or low risk. Moderate risk levels fall between z-scores of 1 and 2, while high risk levels are above a z-score of 2, making z-scores an essential part of evaluating cholesterol risk profiles.