Problem 22 - 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. $$ P(x \geq 1.01) $$

Step-by-Step Solution

Answer

The probability \( P(x \geq 1.01) \) is 0.1562.

Step 1: Identify the problem type

We need to determine the probability of a standard normal random variable (with mean 0 and standard deviation 1) being at least 1.01, which corresponds to finding the probability that \( x \geq 1.01 \). This involves using the standard normal distribution table (Table A).

Step 2: Interpret the standard normal table

The standard normal distribution table provides values for \( P(z \leq a) \), where \( a \) is a z-score. Thus, to find \( P(x \geq 1.01) \), we first need to find \( P(x \leq 1.01) \).

Step 3: Look up the z-score in Table A

Use Table A to find the probability \( P(x \leq 1.01) \). For a z-score of 1.01, the table gives a probability of approximately 0.8438

Step 4: Calculate the required probability

Since \( P(x \geq 1.01) = 1 - P(x \leq 1.01) \), compute \( 1 - 0.8438 = 0.1562 \).

Step 5: Conclusion

Thus, the probability that \( x \geq 1.01 \) is 0.1562.

Key Concepts

Continuous Random Variable

A continuous random variable is a variable that can take any value within a given range. Unlike discrete random variables, which take specific, isolated values, continuous random variables can assume an infinite number of possibilities. In the context of probability distribution, a continuous random variable is often described by a probability density function (PDF). This function shows how probabilities are distributed over the values of the variable. In our specific case, we're dealing with a standard normal distribution, which is a type of continuous probability distribution. This distribution represents how values of a random variable are expected to behave over time. A standard normal distribution has a mean of 0 and a standard deviation of 1, making it a special case of normal distribution. The concept of continuous random variables is crucial when dealing with any form of statistical analysis because it allows us to understand and predict behavior within certain datasets. Using continuous random variables, we can calculate probabilities for ranges of values rather than specific outcomes.

Z-Score

A z-score is a statistical measurement that describes a value's relation to the mean of a group of values. In simpler terms, it tells us how many standard deviations a data point is from the average. In the context of a standard normal distribution, where the mean is 0 and the standard deviation is 1, the z-score directly represents the value. To calculate a z-score, you subtract the mean from the data point and then divide by the standard deviation. The resulting z-score can be positive or negative, indicating whether the value is above or below the mean. For example, a z-score of 1.01 means the value is 1.01 standard deviations above the mean. Understanding z-scores is fundamental in probability and statistics because they standardize different datasets to the same scale. This standardization allows for easy comparison and is preferred when using statistical tables like the standard normal distribution table.

Probability Calculation

Probability calculation in a continuous setting involves determining the likelihood of a range of values occurring within a probability distribution. In standard normal distribution, probabilities are derived using z-scores. The exercise we consider here asks us to find the probability of a standard normal random variable being at least 1.01.To perform such a calculation, we use the cumulative distribution function (CDF), which gives the probability that the variable takes a value less than or equal to a specific z-score. However, we wanted to find the probability that a value is at least 1.01. Thus, we need the complement of the CDF result.We calculate it using the formula:- \( P(x \geq z) = 1 - P(x \leq z) \)- Given: \( P(x \leq 1.01) = 0.8438 \)- So: \( P(x \geq 1.01) = 1 - 0.8438 = 0.1562 \)This result tells us that there's a 15.62% chance a value from our distribution is at least 1.01.

Standard Normal Table

The standard normal table, also known as the Z-table, is a mathematical table that allows users to find the probability of a z-score occurring in a standard normal distribution. It provides the cumulative probability associated with a particular z-score, essentially giving the area under the curve to the left of a z-score on a standard normal distribution graph. When using the standard normal table, you locate the row corresponding to the integer and the first decimal places of your z-score, and then match this with the appropriate column for the second decimal place. This intersection gives the cumulative probability. For the z-score of 1.01 in our exercise, the standard normal table shows a cumulative probability of 0.8438. This represents the total probability from the extreme left of the distribution up to 1.01. To find the probability of values greater than 1.01, you subtract this cumulative probability from 1. The standard normal table simplifies the process of calculating probabilities, making it a vital tool for statisticians and students learning about normal distributions.