Problem 17 - 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. $$ P(1.35 \leq x \leq 1.45) $$

Step-by-Step Solution

Answer

The probability that \(x\) is between 1.35 and 1.45 is 0.0150.

Step 1: Understand the Standard Normal Distribution

The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. Probabilities for this distribution are typically found using tables, often called \( z \)-tables, or standard normal distribution tables.

Step 2: Convert Range to z-scores

For a standard normal distribution, the values \( x = 1.35 \) and \( x = 1.45 \) are already \( z \) scores because the standard normal distribution has a mean of 0 and a standard deviation of 1.

Step 3: Consult the z-table for Probabilities

Find the probability values corresponding to \( z = 1.35 \) and \( z = 1.45 \) using Table A. Typically, these tables give the area from the mean to the \( z \)-value. For \( z = 1.35 \), the table provides an area of 0.4115 and for \( z = 1.45 \), an area of 0.4265.

Step 4: Calculate the Probability for the Interval

To find the probability that \( x \) falls between 1.35 and 1.45, subtract the probability of \( x \leq 1.35 \) from the probability of \( x \leq 1.45 \): \[ P(1.35 \leq x \leq 1.45) = P(x \leq 1.45) - P(x \leq 1.35) = 0.4265 - 0.4115 \] Calculate this difference to find the desired probability.

Step 5: Compute and Conclude

Perform the subtraction: \[ 0.4265 - 0.4115 = 0.0150 \] Thus, the probability that \( x \) lies between 1.35 and 1.45 is 0.0150.

Key Concepts

Probability Calculation

In statistics, calculating probability is about determining how likely an event is to occur. When working with continuous random variables, we often use the standard normal distribution. This helps us in finding probabilities for ranges of values.

To explain the process further, let's consider the example of finding the probability that a variable falls between 1.35 and 1.45 in a standard normal distribution. If you have two endpoints, you begin by finding the probability of the variable being less than or equal to each endpoint. This is indicated as \( P(x \leq a) \) for some number \( a \).

Once you know each of these probabilities, calculate the probability for the interval by subtraction. This means subtracting the probability of the lower endpoint from the probability of the higher endpoint. Hence, probability calculation involves:

• Defining the endpoints of the interval.
• Finding the related probabilities using a tool like a z-table.
• Subtracting to find the probability between the two points.

Let's take our example. If the probability for \( z = 1.45 \) is 0.4265 and for \( z = 1.35 \) is 0.4115, the probability that your variable is between these two points is computed as 0.4265 minus 0.4115, resulting in 0.0150.

z-scores

Understanding z-scores is vital when working with the standard normal distribution. A z-score tells us how many standard deviations a particular value is from the mean. In a standard normal distribution, the mean is 0, and the standard deviation is 1.

Z-scores help standardize different normal distributions allowing for comparisons. To compute a z-score, use the formula:
\[ z = \frac{(X - \mu)}{\sigma} \]
where:

• \( X \) is the value of the random variable.
• \( \mu \) is the mean of the distribution.
• \( \sigma \) is the standard deviation.

However, in a standard normal distribution, values like \( x = 1.35 \) or \( x = 1.45 \) are already z-scores by themselves because the mean is 0 and the standard deviation is 1.

If a z-score is positive, it tells you that your value is above the mean. If it's negative, it's below the mean. Knowing z-scores not only helps in calculating probabilities but is also crucial for interpreting where your data fits relative to other points in your distribution.

z-tables

Z-tables, often known as standard normal distribution tables, are tools used to find probabilities associated with z-scores. When you work with standard normal distributions, these tables are indispensable for probability calculations.

Inside the table, the rows and columns help you find the area to the left of a specific z-score. This area is the cumulative probability, which helps in determining how likely a value is to occur.

For example, if we want to find \( P(x \leq 1.45) \), we look at the z-table for a z-score of 1.45. According to Table A, this might give an area of around 0.4265. This could be interpreted as a 42.65% chance that a z-score of less than or equal to 1.45 will occur.

• Z-tables list the probabilities (or areas) to the left of a given z-score.
• To find probabilities between two z-scores, you subtract the smaller area from the larger one as seen with \( P(x \leq 1.45) - P(x \leq 1.35) \).

Z-tables help transfer standard deviations into probabilities, enabling you to quantify data behavior, essential for statistical analysis.