Problem 27 - Calculus and its applications

Question

Let \(x\) be a continuous random variable that is normally distributed with mean \(\mu=22\) and standard deviation \(\sigma=5 .\) Using Table A, find the following. $$ P(18 \leq x \leq 26) $$

Step-by-Step Solution

Answer

The probability that \(x\) is between 18 and 26 is 0.5762.

Step 1: Understand the Given Problem

We need to find the probability that the continuous random variable \(x\), which is normally distributed with mean \(\mu = 22\) and standard deviation \(\sigma = 5\), falls between 18 and 26, i.e., \(P(18 \leq x \leq 26)\).

Step 2: Convert the Range to Z-Scores

To use Table A (standard normal distribution table), we first convert the values 18 and 26 to their respective Z-scores using the formula: \( Z = \frac{X - \mu}{\sigma} \). For 18: \( Z_{18} = \frac{18 - 22}{5} = -0.8 \). For 26: \( Z_{26} = \frac{26 - 22}{5} = 0.8 \).

Step 3: Use Z-Table to Find Probabilities

Consult the standard normal distribution (Z) table to find the probabilities associated with the Z-scores. \( P(Z \leq 0.8) \) corresponds to approximately 0.7881 and \( P(Z \leq -0.8) \) corresponds to approximately 0.2119.

Step 4: Calculate the Probability for the Range

To find \(P(18 \leq x \leq 26)\), we need the difference between the two probabilities: \(P(18 \leq x \leq 26) = P(Z \leq 0.8) - P(Z \leq -0.8) = 0.7881 - 0.2119 = 0.5762\).

Key Concepts

Continuous Random Variable

A continuous random variable is a variable that can take an infinite number of possible values. These values are often measured and fall within a range across the continuous spectrum. Unlike discrete variables that have specific, countable values, continuous random variables can be any value within an interval.
For example, when we talk about the amount of water in a glass or the exact height of a student, these quantities can be precisely measured down to fractions, making them continuous.
In probability, continuous random variables are important because they allow us to model phenomena in the natural world with precision and flexibility. We can calculate probabilities for ranges of values using probability density functions (PDFs), often represented graphically as curves that show how likely different values are. The area under this curve within a certain interval will give us the probability that our variable will fall within that interval.

Z-Score

The concept of a Z-Score is integral in statistics, particularly when working with normally distributed data. A Z-score signifies how many standard deviations an element is from the mean. It's a way to standardize data, making it easier to compare different data points from distinct normal distributions.
Z-scores are calculated using the formula:\[ Z = \frac{X - \mu}{\sigma} \]where \(X\) is a value from the data set, \(\mu\) is the mean of the data set, and \(\sigma\) is the standard deviation.
A Z-score tells you:

• How far and in what direction (above or below) an element is from the mean.
• If you have a Z-score of 0, it means the data point is exactly at the mean.
• A positive Z-score indicates the value is above the mean, while a negative Z-score indicates it's below.

Using Z-scores allows us to use standard normal distribution tables to find probabilities associated with any normal distribution.

Standard Deviation

The "Standard Deviation" is a key concept in statistics, representing the amount of variation or dispersion in a set of values. Simply put, it tells us how much individual measurements of a dataset typically differ from the mean or average of those measurements.
The standard deviation, denoted as \(\sigma\), is calculated as the square root of the variance. Variance is a measure of how spread out the numbers in a data set are. The formula for standard deviation is:\[ \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (X_i - \mu)^2} \]where \(X_i\) are the data points, \(\mu\) is the mean, and \(N\) is the number of data points.
A small standard deviation means that the values in a dataset are close to the mean, whereas a large standard deviation indicates that the values are spread out over a wider range. In the context of a normal distribution, about 68% of the data falls within one standard deviation of the mean, and 95% falls within two standard deviations.

Probability Calculation

Probability Calculation is a critical aspect of statistics, where the goal is to find out how likely events are to occur. When dealing with a normal distribution, this often involves converting raw scores to Z-scores and using them to determine probabilities.
Here's how it typically works:

• Identify the probability question, such as finding the likelihood that a variable will fall between two points.
• Transform those values to Z-scores using the formula \( Z = \frac{X - \mu}{\sigma} \)
• Consult a Z-table, which provides the probability that a standard normal random variable falls below a particular Z-score.
• Subtract probabilities when looking for the probability that a variable falls between two Z-scores.

For example, if you want to find \(P(18 \leq x \leq 26)\), convert the scores to Z-scores and use the Z-table to find their probabilities. Then, subtract the probability of the lower Z-score from the probability of the higher Z-score to get your answer.This method provides a clear and precise way to perform probability calculations for data involving normal distributions.