Problem 20 - Calculus and its applications

Question

Let \(x\) be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. $$ P(-1.27 \leq x \leq-0.58) $$

Step-by-Step Solution

Answer

The probability is approximately 0.1790.

Step 1: Understanding the Problem

We need to find the probability that a standard normal variable \( x \) lies between \(-1.27\) and \(-0.58\). This involves using the standard normal distribution table (Table A) to find the corresponding probabilities.

Step 2: Using the Standard Normal Table

The standard normal table provides the cumulative probability \(P(Z \leq z)\) for a standard normal variable. Locate \(-1.27\) in the table to find \(P(Z \leq -1.27)\) and \(-0.58\) to find \(P(Z \leq -0.58)\).

Step 3: Find Cumulative Probabilities

From the table, \(P(Z \leq -1.27) \approx 0.1020\) and \(P(Z \leq -0.58) \approx 0.2810\). These values represent the probabilities of \(Z\) being less than or equal to \(-1.27\) and \(-0.58\), respectively.

Step 4: Calculate the Between Probability

Calculate \(P(-1.27 \leq x \leq -0.58)\) by subtracting the cumulative probability of \(-1.27\) from that of \(-0.58\): \(0.2810 - 0.1020 = 0.1790\).

Key Concepts

Probability Calculation

The process of probability calculation in the context of the standard normal distribution involves determining the likelihood that a random variable falls within a certain interval. This probability is a key concept in statistics and is often represented by an area under the normal curve. In our specific exercise, we're interested in calculating the probability that a standard normal variable \( x \) falls between two values, \(-1.27\) and \(-0.58\).

Probability calculation typically follows these steps:

• Identify the range of the variable for which you want to calculate the probability, in this case, \(-1.27\) to \(-0.58\).
• Use relevant statistical tables or computational tools to compute the probability for these ranges.
• The final probability is found by subtracting the smaller cumulative probability from the larger one (i.e., \( P(-0.58) - P(-1.27) \)).

This subtraction gives the probability that the variable falls between the specified limits, illustrating a fundamental approach to tackling probability in statistics.

Cumulative Probability

Cumulative probability is crucial in understanding how frequently certain outcomes occur in a distribution. In a standard normal distribution, which has a mean of 0 and a standard deviation of 1, cumulative probability is used to determine the probability that a random variable \( Z \) is less than or equal to a specific value.

Here's how cumulative probability works in this situation:

• The cumulative probability function provides the area under the curve to the left of a given \( Z \) value.
• For \( Z = -1.27 \), the cumulative probability \( P(Z \leq -1.27) \) is the probability that \( Z \) is less than or equal to \(-1.27\), and it equates to about 0.1020 based on our standard normal table.
• Similarly, \( P(Z \leq -0.58) \approx 0.2810 \). This value is larger, reflecting the additional area covered under the curve from \(-1.27\) to \(-0.58\).

Thus, cumulative probability aids in understanding the spread and likelihood of various outcomes in a normal distribution.

Using Standard Normal Table

The standard normal table, also known as the Z-table, is an essential tool used to find probabilities associated with the standard normal distribution. It tabulates cumulative probabilities, which help in determining the proportion of values that are less than a specified Z-score in a normal distribution.

Using the table involves a straightforward approach:

• Locate the Z-score of the interest along the rows and columns of the table.
• The table entry corresponding to the Z-score provides the cumulative probability of the standard normal random variable being less than or equal to that Z-score.
• For our exercise, you found \( P(Z \leq -1.27) \approx 0.1020 \) and \( P(Z \leq -0.58) \approx 0.2810 \).

This demonstrates how the Z-table simplifies the task of finding probabilities for various intervals, making it a valuable resource for statistical analysis and probability calculations. Remember, Z-tables standardize probabilities by assuming a mean of 0 and standard deviation of 1, making them versatile for many applications.