Problem 13
Question
Let \(x\) be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. $$ P(-1.37 \leq x \leq 0) $$
Step-by-Step Solution
Verified Answer
The probability \(P(-1.37 \leq x \leq 0)\) is approximately 0.4147.
1Step 1: Understanding the Problem
We need to find the probability that the standard normal random variable \(x\) falls between -1.37 and 0, i.e., \(P(-1.37 \leq x \leq 0)\). This involves using the standard normal distribution table (often called Table A).
2Step 2: Use Table A for Upper Limit 0
Since we are dealing with a standard normal distribution, the probability \(P(x \leq 0)\) is found directly from the table. The value at 0 for the standard normal distribution is 0.5. Thus, \(P(x \leq 0) = 0.5\).
3Step 3: Use Table A for Lower Limit -1.37
Look up the value \(-1.37\) in the standard normal distribution table. This gives \(P(x \leq -1.37)\). According to Table A, \(P(x \leq -1.37) \approx 0.0853\).
4Step 4: Calculate the Desired Probability
The probability that \(x\) is between -1.37 and 0 is given by subtracting \(P(x \leq -1.37)\) from \(P(x \leq 0)\): \[ P(-1.37 \leq x \leq 0) = P(x \leq 0) - P(x \leq -1.37) = 0.5 - 0.0853 = 0.4147. \]
5Step 5: Conclusion
The probability that the standard normal random variable \(x\) falls between -1.37 and 0 is approximately 0.4147.
Key Concepts
Continuous Random VariableStandard Normal Distribution TableProbability Calculation
Continuous Random Variable
A continuous random variable is a variable that can take an infinite number of possible values. Unlike a discrete random variable, which takes on fixed values (like rolling a die), a continuous random variable can be any real number within a given range.
In the context of the standard normal distribution, which is a key concept in statistics, the random variable often represents a measure like height, weight, or, as in the original exercise, the variable \(x\). Continuous random variables are depicted on a continuous probability distribution, with the area under the curve representing probabilities.
This area is crucial because the probability of the variable falling within any specific range can be found by calculating the area under the curve between those two points. Thus, understanding continuous random variables helps us tackle complex statistical problems like the one in the exercise.
In the context of the standard normal distribution, which is a key concept in statistics, the random variable often represents a measure like height, weight, or, as in the original exercise, the variable \(x\). Continuous random variables are depicted on a continuous probability distribution, with the area under the curve representing probabilities.
This area is crucial because the probability of the variable falling within any specific range can be found by calculating the area under the curve between those two points. Thus, understanding continuous random variables helps us tackle complex statistical problems like the one in the exercise.
Standard Normal Distribution Table
The standard normal distribution table, often referred to as Table A, is a tool used to find probabilities for a standard normal distribution, specifically where \(x\) follows a normal distribution with a mean of 0 and a standard deviation of 1.
This table lists values and their corresponding cumulative probabilities. The table is symmetric around zero, so it only includes positive \(z\)-scores, with the understanding that you can use this symmetry to find the probabilities associated with negative \(z\)-scores.
This table lists values and their corresponding cumulative probabilities. The table is symmetric around zero, so it only includes positive \(z\)-scores, with the understanding that you can use this symmetry to find the probabilities associated with negative \(z\)-scores.
- To find the probability of \(x\) being less than a certain value: Locate the \(z\)-score in the table to find the cumulative probability.
- This can directly provide the probability for one of the limits, as shown in the solution with \(P(x \leq 0) = 0.5\).
- For negative values, such as \(-1.37\), you must use the table's symmetry to deduce that \(P(x \leq -1.37)\) is the same as 1 minus the probability for positive \(1.37\). However, many tables already include negative values for quick reference.
Probability Calculation
Probability calculation is a vital process in statistics where you determine the likelihood of a random variable falling within a certain range. In problems involving the standard normal distribution, this often involves subtracting probabilities from each other to find the desired range.
In the given example, you are asked to find \(P(-1.37 \leq x \leq 0)\). By using the standard normal distribution table and cumulative probabilities, you can compute this range.
In the given example, you are asked to find \(P(-1.37 \leq x \leq 0)\). By using the standard normal distribution table and cumulative probabilities, you can compute this range.
- First, determine \(P(x \leq 0)\), directly from the table, as it is straightforwardly given as 0.5.
- Next, find \(P(x \leq -1.37)\) from the table, which in the example is 0.0853.
- Finally, calculate the needed probability by subtracting the latter from the former: \(P(-1.37 \leq x \leq 0) = 0.5 - 0.0853 = 0.4147\).
Other exercises in this chapter
Problem 12
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