Problem 16 - 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. $$ [\mu-2 \sigma, \mu+\sigma] $$

Step-by-Step Solution

Answer

About 81.85% of the population falls within the interval \([\mu-2\sigma, \mu+\sigma]\).

Step 1: Understanding the Problem

We need to determine the fraction of the population that falls within the interval \([\mu-2\sigma, \mu+\sigma]\) of a normally distributed character. This requires evaluating the probability of this range using the properties of normal distribution.

Step 2: Calculate the Z-Scores

Firstly, compute the z-scores for the boundaries of the interval relative to the mean \(\mu\). The z-score formula is \(Z = \frac{X - \mu}{\sigma}\). For \(X = \mu - 2\sigma\), the z-score is \(Z_1 = \frac{\mu - 2\sigma - \mu}{\sigma} = -2\). Similarly, for \(X = \mu + \sigma\), the z-score is \(Z_2 = \frac{\mu + \sigma - \mu}{\sigma} = 1\).

Step 3: Use the Normal Distribution Table

With z-scores of \(-2\) and \(1\), refer to the standard normal distribution table (or use a calculator) to find the corresponding probabilities. For \(Z = -2\), the cumulative probability is approximately \(0.0228\). For \(Z = 1\), the cumulative probability is approximately \(0.8413\).

Step 4: Calculate the Probability of the Interval

The probability of the interval \([\mu-2\sigma, \mu+\sigma]\) is the difference between the cumulative probabilities at these points: \(P(-2 < Z < 1) = P(Z < 1) - P(Z < -2) = 0.8413 - 0.0228 = 0.8185\).

Step 5: Conclusion

Thus, approximately \(81.85\%\) of the population falls within this given interval of the normal distribution.

Key Concepts

Z-Scores

When dealing with normal distributions, z-scores are extremely useful. They standardize data points by converting them into the number of standard deviations away from the mean. This makes comparison straightforward and allows us to use standard tables for probability analysis.

The formula for computing 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 distribution. By converting data into z-scores, any normal distribution can refer back to the standard normal, which has a mean of 0 and a standard deviation of 1.

For example, if you compute the z-score for the boundaries \(X = \mu - 2\sigma\) and \(X = \mu + \sigma\) from the problem, you get \(Z_1 = -2\) and \(Z_2 = 1\). These translate the original distribution into a standardized format.

Cumulative Probability

Cumulative probability refers to the probability that a random variable is less than or equal to a particular value in a given distribution. It is the area under the probability distribution curve from the leftmost point up to a specific x-value.

We calculate the cumulative probability using cumulative distribution functions (CDF). For a z-score \(Z\), the CDF gives the probability that a standard normal variable is less than \(Z\). This makes z-scores very handy since we can easily find cumulative probabilities using a standard normal table.

• For \(Z = -2\), the cumulative probability is approximately \(0.0228\).
• For \(Z = 1\), it is approximately \(0.8413\).

The difference between these cumulative probabilities gives the probability that a random variable falls within the given interval, which is the essence of interval probabilities in a normal distribution. In this case, it is \(0.8413 - 0.0228 = 0.8185\), meaning 81.85% of values fall within the interval.

Quantitative Character

A quantitative character is a trait or attribute that can be measured and expressed numerically. These are often represented with continuous data, like height, weight, or temperature.

These characters usually follow a specific type of probability distribution, with the normal distribution being the most common. In a normal distribution, data is symmetrically distributed around the mean \(\mu\), with most points clustering within a few standard deviations \(\sigma\).

When dealing with normally distributed quantitative characters, the properties of the normal curve facilitate various analyses.

• These include determining intervals where data points are likely to fall.
• Assessing probabilities that a value is within a specified range.

Understanding these concepts is crucial when analyzing how a quantitative character behaves across a population, allowing predictions and better comprehension of observed data patterns.