Problem 29 - Calculus for Biology and Medicine

Question

Suppose that \(X\) is normally distributed with mean 2 and standard deviation 1. Find \(P(0 \leq X \leq 3)\).

Step-by-Step Solution

Answer

The probability that \(X\) is between 0 and 3 is 0.8185.

Step 1: Standardize the Distribution

Given that \(X\) is normally distributed with a mean \(\mu = 2\) and standard deviation \(\sigma = 1\), we first convert \(X\) to a standard normal distribution. This involves finding the standard score or \(z\)-score for \(X = 0\) and \(X = 3\) using the formula: \[ z = \frac{X - \mu}{\sigma} \] For \(X = 0\), \[ z_0 = \frac{0 - 2}{1} = -2 \]For \(X = 3\), \[ z_3 = \frac{3 - 2}{1} = 1 \]

Step 2: Use the Standard Normal Distribution Table

Next, we find the probabilities corresponding to \(z_0\) and \(z_3\) using a standard normal distribution table (Z-table).For \(z = -2\), \(P(Z \leq -2) = 0.0228\).For \(z = 1\), \(P(Z \leq 1) = 0.8413\).

Step 3: Calculate the Probability

The probability that \(X\) falls between 0 and 3 is given by the difference in the cumulative probabilities obtained from the Z-table.\[ P(0 \leq X \leq 3) = P(Z \leq 1) - P(Z \leq -2) = 0.8413 - 0.0228 \] So,\[ P(0 \leq X \leq 3) = 0.8185 \]

Key Concepts

Standard Normal Distribution

Understanding the concept of a standard normal distribution is crucial for working with probability problems linked to normally distributed random variables. A standard normal distribution is a special case of the normal distribution where the mean (bc) is 0 and the standard deviation (c3) is 1. This standardization simplifies the comparison of different datasets.
When a random variable follows a standard normal distribution, it's denoted by the letter Z, and its associated probabilities can be determined using a Z-table.

• It helps in converting any normal distribution into a standard form.
• It provides a baseline for comparing variances of datasets.

By transforming a normally distributed variable (like X in our exercise) into a standard normal variable (Z), you can easily use the Z-table to find probabilities for any segment of the distribution.

Z-Score

A Z-score is the number of standard deviations a data point, or in this context, a value of X, is from the mean (bc) of the distribution. It is calculated using the formula:
\[ z = \frac{X - \mu}{\sigma} \]
This conversion facilitates the use of a standard normal distribution to evaluate probabilities for a dataset with any mean and standard deviation.
In the example, we calculated Z-scores for X values within the interval [0, 3]. For example, for X = 0, the Z-score is \(-2\), indicating that it is 2 standard deviations below the mean of the distribution.

• A Z-score lets you "standardize" a normal distribution into the well-defined standard normal distribution.
• With Z-scores, complexities due to differing means and standard deviations are eliminated.

By grasping Z-scores, you can convert ordinary data points into a simplified form that represents their position relative to the mean.

Probability Calculation

With Z-scores at hand, you need to calculate the probability of a certain event within a distribution. The Z-score allows you to find the probability of a value lying to the left of it using a Z-table. In our exercise, after calculating the Z-scores for X = 0 and X = 3, we used the Z-table to determine their respective probabilities.

• Z = -2, corresponding to a probability of 0.0228.
• Z = 1, corresponding to a probability of 0.8413.

The end goal of calculating these probabilities is to determine the likelihood of the random variable falling between the two specified points. This method proves very effective in finding probabilities related to any normal distribution, regardless of its specific mean or standard deviation.

Cumulative Probability

Cumulative probability represents the likelihood that a random variable will be less than or equal to a particular value. It punches in the probability of all possible outcomes up to that point, hence 'cumulative'.
In our exercise, we determined the cumulative probabilities for Z = -2 and Z = 1, which helped us conclude that the probability of X lying between 0 and 3 is the difference between these cumulative probabilities. Mathematically, it was calculated as:
\[ P(0 \leq X \leq 3) = P(Z \leq 1) - P(Z \leq -2) = 0.8185 \]
This step involves deducing the probability of a range by subtracting cumulative probabilities, which is vital when calculating probabilities over intervals that do not start from the beginning of the distribution.

• Cumulative probabilities leverage the Z-table to give the full probability up to any point.
• It's particularly useful when evaluating the probability of an event occurrence within a specific range.

When mastered, cumulative probability calculations become a handy tool for anyone operating within statistical boundaries.