Problem 30 - 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. $$ P(17.2 \leq x \leq 21.7) $$

Step-by-Step Solution

Answer

The probability is approximately 0.3076.

Step 1: Identify the Problem

We need to find the probability that the continuous random variable \(x\) falls between 17.2 and 21.7, given that \(x\) is normally distributed with \(\mu = 22\) and \(\sigma = 5\).

Step 2: Standardize the Random Variable

Convert the given \(x\) values (17.2 and 21.7) to standard normal \(z\) scores using the formula \(z = \frac{x - \mu}{\sigma}\). For \(x = 17.2\), the \(z\) score is \(z_1 = \frac{17.2 - 22}{5} = -0.96\). For \(x = 21.7\), the \(z\) score is \(z_2 = \frac{21.7 - 22}{5} = -0.06\).

Step 3: Locate Values in the Z-Table

Using a standard normal distribution table (Table A), locate the probabilities associated with \(z_1 = -0.96\) and \(z_2 = -0.06\). The table provides the probability that \(z\) is less than a given value. For \(z_1 = -0.96\), find \(P(Z < -0.96)\), and for \(z_2 = -0.06\), find \(P(Z < -0.06)\).

Step 4: Calculate Probability Between z-Scores

Use the probabilities from the Z-table to calculate \(P(17.2 \leq x \leq 21.7)\). Subtract the probability associated with \(z_1\) from the probability associated with \(z_2\): \(P(-0.96 < Z < -0.06) = P(Z < -0.06) - P(Z < -0.96)\).

Step 5: Find the Final Probability

Assume from Table A that \(P(Z < -0.96) = 0.1685\) and \(P(Z < -0.06) = 0.4761\). Therefore, \(P(17.2 \leq x \leq 21.7) = 0.4761 - 0.1685 = 0.3076\).

Key Concepts

Probability Calculation

Probability calculation in a normal distribution is all about finding the likelihood that a certain value falls within a specified range. This is like predicting the chance of an event occurring within a particular segment of data. By understanding the shape and properties of a normal distribution, particularly the bell curve, you can estimate these probabilities.
The key steps include:

• Identifying the mean (bc) and standard deviation () given in the problem. These values tell you where the center of the distribution lies and how spread out the data is.
• Standardizing the values using the Z-score formula, which helps you compare the values of interest to the mean.
• Using a Z-table to find the relative probabilities for each Z-score, which show you the likelihood that a value is less than a specific point.

By breaking down the problem into these steps, probability calculation becomes a manageable process that helps you make data-driven decisions.

Standard Normal Distribution

The standard normal distribution is a special form of the normal distribution, standardized to have a mean (bc) of 0 and a standard deviation (f) of 1. This standardization allows for simplicity when calculating probabilities and comparing values across different normal distributions.
A standard normal distribution is characterized by:

• A perfectly symmetrical bell shape, centered at zero.
• The total area under the curve equals 1, representing the whole probability space.
• Values along the x-axis being in terms of Z-scores, which show how many standard deviations an element is from the mean.

Using the standard normal distribution simplifies calculations because you can refer to standardized Z-tables without adjusting for different means and standard deviations. This makes it easier to analyze data that initially might have varied scales, by converting them into the common frame of reference the standard normal offers.

Z-Score Conversion

The Z-score conversion process helps to transform your original data into a standard format that aligns with the standard normal distribution. This process essentially tells you how far, and in what direction, a data point is from the mean, measured in terms of standard deviations.
To convert a raw score to a Z-score, use the formula:\[ z = \frac{x - \mu}{\sigma} \]where \(x\) is the raw score, \(\mu\) is the mean, and \( \sigma \) is the standard deviation.

• If the Z-score is positive, that means the data point is above the mean.
• If it's negative, the data point is below the mean.

Converting to a Z-score allows you to leverage the Z-table to find probabilities and percentiles for any given score. This approach is essential in data analysis because it creates a universal metric for different data. This standardization facilitates meaningful comparisons and inferences.