Problem 19 - 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. $$ (-\infty, \mu-2 \sigma] $$

Step-by-Step Solution

Answer

Approximately 2.28% of the population falls within the interval \((-\infty, \mu - 2\sigma]\).

Step 1: Understand the Normal Distribution

The normal distribution is a continuous probability distribution characterized by a symmetric bell-shaped curve. A key characteristic of this distribution is that it is defined by two parameters: the mean \( \mu \) and the standard deviation \( \sigma \). The mean determines the center of the distribution, while the standard deviation determines the spread of the distribution.

Step 2: Definition of the Interval

The interval \( (-\infty, \mu - 2\sigma] \) is the range from negative infinity up to and including the point \( \mu - 2\sigma \). This means we are interested in the portion of the distribution that falls below \( \mu - 2\sigma \).

Step 3: Standard Normal Distribution Conversion

Convert the cutoff point \( \mu - 2\sigma \) into a standard normal variable \( z \) using the formula: \( z = \frac{x - \mu}{\sigma} \). For our interval, \( z = \frac{(\mu - 2\sigma) - \mu}{\sigma} = \frac{-2\sigma}{\sigma} = -2 \).

Step 4: Use the Standard Normal Distribution Table

The standard normal distribution table (or \( z \)-table) provides the area (or probability) for values below a specific \( z \)-score. For \( z = -2 \), look up the corresponding cumulative probability, which is approximately 0.0228. This value represents the area or probability of observations falling to the left of \( \mu - 2\sigma \).

Step 5: Interpret the Result

The probability of falling below \( \mu - 2\sigma \) is approximately 0.0228. This means about 2.28% of the population falls within the given interval \( (-\infty, \mu - 2\sigma] \).

Key Concepts

Standard Deviation

The standard deviation, denoted as \( \sigma \), is a fundamental concept in statistics and is essential for understanding normal distributions. It measures the amount of variation or dispersion in a set of values. When values are spread out more widely from the mean, the standard deviation is larger. Conversely, when values are closely clustered around the mean, the standard deviation is smaller.

In a normal distribution, the standard deviation is crucial because it allows for the determination of specific intervals relevant to the mean. These intervals help predict the probability of data falling within certain ranges, all revolving around the mean. Specifically, in a normal distribution:

• Approximately 68% of values lie within one standard deviation of the mean (\( \mu \pm \sigma \)).
• About 95% fall within two standard deviations (\( \mu \pm 2\sigma \)).
• Almost 99.7% are within three standard deviations (\( \mu \pm 3\sigma \)).

Understanding how the standard deviation influences the shape of the normal curve is key to calculating probabilities for specific data ranges, like in the given exercise, where the focus is on \( \mu - 2\sigma \).

Z-Score

A z-score, or standard score, is a way of describing a data point in terms of how many standard deviations it is from the mean. By converting data points into z-scores, any normal distribution can be transformed into the standard normal distribution, which has a mean of 0 and a standard deviation of 1.

The formula for calculating a z-score is:\[ z = \frac{x - \mu}{\sigma} \] where \( x \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation of the population.

In the exercise provided, the cutoff point \( \mu - 2\sigma \) was converted into the z-score \( z = -2 \). This step is important because it allows us to use the standard normal distribution table (or z-table) to find the cumulative probability corresponding to this z-score. This table makes it easy to find probabilities without performing complex calculus, since all normal distributions can be related to the standard one.

Cumulative Probability

Cumulative probability refers to the probability that a random variable will have a value less than or equal to a specific value. For the standard normal distribution, cumulative probability tells us the likelihood that a z-score is lower than a given threshold.

In the context of our exercise, we were interested in the cumulative probability up to \( \mu - 2\sigma \) in a normal distribution. This involves looking up the probability up to the z-score of \( -2 \) using a z-table, which gives us the cumulative probability of 0.0228.

• This value means 2.28% of observations are less than \( \mu - 2\sigma \).
• The ability to determine this fraction of the population helps in various fields, such as quality control and finance, by predicting outcomes.

Using cumulative probabilities, one can make informed decisions based on the likelihood of different scenarios, making it a powerful tool for statistical analysis and interpretation.