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Magazine Chapter 10: Problem 4
Let \(Z\) be a Normal \((0,1)\) random variable. Find the probability that \(Z\) is in the interval. $$ [-1.29,2.04] $$
Short Answer
Expert verified
0.8808
Step by step solution
01
- Understand the Normal Distribution
Recognize that the normal distribution with mean \( \mu = 0\ \) and standard deviation \(\ \sigma = 1\ \) is the standard normal distribution.
02
- Use the Standard Normal Table
Consult the standard normal (Z) table to find the cumulative probabilities for both \(\ z = -1.29\ \) and \(\ z = 2.04\ \).
03
- Find Cumulative Probabilities
From the Z table, locate the cumulative probability at \(-1.29\), which corresponds to \ P(Z \leq -1.29) = 0.0985 \). Similarly, find the probability at \(2.04 \), giving \( P(Z \leq 2.04) = 0.9793 \.
04
- Calculate the Desired Probability
Compute the probability that \(Z\ \) lies within the interval \([-1.29, 2.04]\) by subtracting the cumulative probability at \(-1.29\) from that at \(2.04\): \( P(-1.29 \ \leq Z \ \leq 2.04) = P(Z \ \leq 2.04) - P(Z \ \leq -1.29) = 0.9793 - 0.0985 \).
05
- Simplify the Result
Simplify the result to find the final probability: \(0.9793 - 0.0985 = 0.8808\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
standard normal distribution
The standard normal distribution plays a crucial role in probability and statistics. It represents a normal distribution with a mean \( \mu = 0 \) and a standard deviation \( \sigma = 1 \). This distribution is symmetric around zero, meaning that the mean, median, and mode all coincide at the center. Because of its standardized form, we can easily use tables (known as Z tables) to find probabilities and percentiles.
Some key properties of the standard normal distribution include:
Some key properties of the standard normal distribution include:
- The total area under the curve equals 1.
- About 68% of the data falls within one standard deviation (between -1 and 1).
- Approximately 95% fall within two standard deviations.
- Almost 99.7% of the data lies within three standard deviations.
cumulative probability
Cumulative probability is the likelihood that a random variable is less than or equal to a given value. In the context of a standard normal distribution, it helps in finding the area under the curve to the left of a specific Z score.
To get the cumulative probability from a Z table:
To get the cumulative probability from a Z table:
- Locate the row corresponding to the first two digits of your Z score.
- Find the column for the second decimal place of your Z score.
- The cell where the row and column intersect gives you the cumulative probability.
Z table calculation
The Z table is a crucial tool for finding cumulative probabilities for standard normal distributions. Here's a step-by-step guide to using a Z table to solve problems like the one in our original exercise:
Step 1:
Step 1:
- Obtain the Z scores for the values you're investigating. In our case, these are -1.29 and 2.04.
- Look up these Z scores in the Z table. For \( Z = -1.29 \), the cumulative probability is \( P(Z \leq -1.29) = 0.0985 \). For \( Z = 2.04 \), it's \( P(Z \leq 2.04) = 0.9793 \).
- Calculate the desired probability by subtracting the smaller cumulative probability from the larger one. This gives the probability that Z lies between the two scores: \( P(-1.29 \leq Z \leq 2.04) = 0.9793 - 0.0985 = 0.8808 \).
