Problem 20 - Calculus for Biology and Medicine

Question

Assume that a quantitative character is normally distributed with mean \(\mu\) and standard deviation \(\sigma .\) Determine what fraction of the population falls into the given interval. $$ [\mu-3 \sigma, \mu] $$

Step-by-Step Solution

Answer

Approximately 49.87% of the population falls within the interval \([\mu - 3\sigma, \mu]\).

Step 1: Understanding the Problem

We are given a normal distribution with mean \( \mu \) and standard deviation \( \sigma \). We need to find out what fraction of the population lies within the interval \([\mu - 3\sigma, \mu]\).

Step 2: Visualizing the Normal Distribution

A normal distribution is symmetric about its mean. For this problem, imagine the bell curve centered at \( \mu \). The values extend towards \( \pm \infty \), but most are within a few standard deviations \( \sigma \) from the mean \( \mu \).

Step 3: Converting to Standard Normal Variable

To solve this problem, we convert the interval limits into \( z \)-scores using the formula: \( z = \frac{X - \mu}{\sigma} \). For \( X = \mu - 3\sigma \), the \( z \)-score is \( z = \frac{(\mu - 3\sigma) - \mu}{\sigma} = -3 \). For the upper limit, \( X = \mu \), the \( z \)-score is \( z = \frac{\mu - \mu}{\sigma} = 0 \).

Step 4: Using the Standard Normal Distribution Table

Now we use the standard normal distribution table (z-table) to find the probability that \( z \) lies between \(-3\) and \(0\). The probability of \( z \leq 0 \) is 0.5. The probability of \( z \leq -3 \) is approximately 0.0013.

Step 5: Calculating the Fraction of the Population

The fraction of the population that lies between \(-3\) and \(0\) is the difference between these probabilities: \( 0.5 - 0.0013 = 0.4987 \).

Step 6: Conclusion

Therefore, approximately 49.87% of the population falls within the interval \([\mu - 3\sigma, \mu]\).

Key Concepts

Mean and Standard Deviation

In statistics, the mean and standard deviation are fundamental concepts that help describe the properties of a dataset, particularly when dealing with a normal distribution. The mean, often symbolized by \( \mu \), is the average of all the data points within a dataset. It is calculated by summing up all the values and then dividing by the number of values. This measure provides us with the central location of the data.
The standard deviation, on the other hand, is expressed as \( \sigma \). It measures the amount of variation or dispersion in the dataset. A low standard deviation means that the data points are close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values. In the context of a normal distribution, knowing both the mean and standard deviation allows us to understand how the data is spread around the mean.
For example, within a normal distribution:

• About 68% of data falls within one standard deviation (\( \pm \sigma \)) of the mean.
• About 95% falls within two standard deviations (\( \pm 2\sigma \)) of the mean.
• And about 99.7% falls within three standard deviations (\( \pm 3\sigma \)) of the mean.

Understanding these percentages highlights how predictable a normal distribution can be when you know the mean and standard deviation.

Z-Score Calculation

Z-score calculation is a way to quantify the number of standard deviations a data point is from the mean of the dataset. To find the z-score for a particular data point \( X \), the formula used is:\[z = \frac{X - \mu}{\sigma}\]This transformation from \( X \) to \( z \) is particularly useful because it converts values from any normal distribution to the standard normal distribution.

• The mean of the standard normal distribution is \( 0 \).
• The standard deviation is \( 1 \).

The z-score helps in understanding how unusual or usual a data point is within the context of the standard normal distribution. If a z-score is negative, the data point is below the mean; if positive, it is above the mean.In the example from the exercise, the interval \([\mu - 3\sigma, \mu]\) translates to z-scores of

• \( z = -3 \)
• \( z = 0 \)

This signifies that we are examining the portion of the population that falls three standard deviations below the mean right up to the mean itself.

Population Fraction

The concept of a population fraction is related to probability and normal distribution. It involves determining what portion of a population falls within a particular interval of interest. In a normally distributed dataset, this is accomplished easily through the use of z-scores and the standard normal distribution table, often called a z-table.
The z-table helps to find probabilities associated with standard normal random variables. These probabilities represent the population fraction we are interested in. In the exercise, we converted the interval \([\mu - 3\sigma, \mu]\) to z-scores and then used a z-table to find the probabilities:

• The probability of \( z \leq 0 \) (all data below the mean) is 0.5.
• The probability of \( z \leq -3 \) is approximately 0.0013.

Therefore, the fraction of the population that lies within the interval \( [\mu - 3\sigma, \mu] \) is calculated as the difference between these probabilities: \[0.5 - 0.0013 = 0.4987\]This means approximately 49.87% of the data falls within the specified range. This calculation shows how the normal distribution's characteristics allow for precise probability estimations and population fraction calculations.