Problem 31 - 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. $$ P(x \geq 20) $$

Step-by-Step Solution

Answer

The probability that \(x\) is greater than or equal to 20 is approximately 0.6554.

Step 1: Understand the Problem

We need to find the probability that a random variable, which is normally distributed with a mean \(\mu = 22\) and standard deviation \(\sigma = 5\), takes a value greater than or equal to 20.

Step 2: Standardize the Variable

Convert the variable to a standard normal variable \(Z\). The formula to convert \(x\) to \(z\) is \( z = \frac{x - \mu}{\sigma} \). So, for \(x = 20\), \( z = \frac{20 - 22}{5} = \frac{-2}{5} = -0.4 \).

Step 3: Use Standard Normal Distribution Table

Look up the Z-table (Table A) to find \( P(Z \geq -0.4) \). Tables typically provide \( P(Z \leq z) \), so we find \( P(Z \leq -0.4) \) first. Looking up the value \(-0.4\), we find \( P(Z \leq -0.4) = 0.3446 \).

Step 4: Calculate the Desired Probability

Since \( P(Z \geq -0.4) = 1 - P(Z \leq -0.4) \), calculate \( P(X \geq 20) = 1 - 0.3446 = 0.6554 \).

Key Concepts

Standard Normal Distribution

The concept of a **Standard Normal Distribution** is a special case of the normal distribution. A standard normal distribution is a continuous probability distribution that follows a normal pattern but is centered around a mean of 0 and has a standard deviation of 1.
Here's what makes it so special and practical:

• It's symmetric around the mean.
• The total area under the curve equals 1, representing a whole probability.
• Approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three, following the empirical rule.

In many cases, converting a normal distribution to a standard normal distribution allows us to use Z-tables to find probabilities and percentiles. This transformation not only simplifies calculations but also enhances understanding by comparing it to a common standard.

Z-score Calculation

A **Z-score Calculation** is key when working with data following a normal distribution. The Z-score is a measure of how many standard deviations an element is from the mean. By converting a value from a normal distribution to a Z-score, we can understand its position relative to the mean.
The process:

• Identify the mean (
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  3) and the standard deviation (
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  3) of the distribution.
• Using the formula: \( z = \frac{x - \mu}{\sigma} \), calculate the Z-score. Here, \( \mu \) is the mean, \( \sigma \) is the standard deviation, and \( x \) is the value to convert.
• An example conversion would result in a Z-score that indicates how far 20 is from the average of 22 given a standard deviation of 5.

Z-scores are valuable because they allow comparison across different normal distributions, standardizing individual scores across different datasets. Understanding this concept is essential for performing probability calculations using standard normal distribution tables.

Probability Calculation

**Probability Calculation** is a crucial step in determining the likelihood that a continuous random variable falls within a certain range in a distribution. Once you calculate a Z-score, the next step involves reading the standard normal distribution table (often called the Z-table).
Steps to perform a probability calculation:

• Look up the Z-score value in the Z-table to find a probability value, \( P(Z \leq z) \).
• The Z-table provides cumulative probabilities from the left, so these values tell us the proportion of values below a given Z-score.
• If you need the probability that a value is greater than a certain point, like in our example of \( P(x \geq 20) \), you'll calculate it as \( 1 - P(Z \leq -0.4) \), giving you the area to the right of that Z-score.

Understanding how to compute probabilities using these tables lays the groundwork for more advanced statistical analyses and interpretation of results. This process turns complex, real-world data into comprehensible probabilities, enhancing decision-making based on statistical findings.