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Magazine Chapter 6: Problem 30
Find the probabilities for each, using the standard normal distribution.
$$ P(-1.23
Short Answer
Expert verified
The probability is 0.3907.
Step by step solution
01
Understand the Z-Scores
The problem asks us to calculate the probability that the standard normal random variable \( z \) lies between \(-1.23\) and 0. This means we need to find the probability for the interval \(-1.23 < z < 0\).
02
Use Standard Normal Distribution Table
Look up the cumulative probabilities for \( z = -1.23 \) and \( z = 0 \) in the standard normal distribution table or use a calculator capable of computing these values. The value for \( z = -1.23 \) is approximately 0.1093, and for \( z = 0 \), the probability is 0.5.
03
Calculate the Required Probability
To find \( P(-1.23 < z < 0) \), subtract the cumulative probability for \( z = -1.23 \) from the cumulative probability for \( z = 0 \). This is calculated as \( P(-1.23 < z < 0) = P(z < 0) - P(z < -1.23) = 0.5 - 0.1093 \).
04
Solve the Subtraction
Perform the calculation: \( 0.5 - 0.1093 = 0.3907 \). Thus, the probability that \(-1.23 < z < 0\) is 0.3907.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Z-Scores
Z-scores are a way to measure how many standard deviations an element is from the mean. They help in determining the position of a particular value within a normal distribution. Think of them as a standardized measure, making it easier to compare different data points across different distributions.
- A Z-score of 0 indicates the value is exactly at the mean.
- Positive Z-scores indicate values above the mean.
- Negative Z-scores indicate values below the mean.
The Role of Cumulative Probability
Cumulative probability represents the likelihood that a random variable falls within a specified range or below a specified point. For the standard normal distribution, it shows the probability that a value is less than or equal to a particular Z-score.
Understanding cumulative probability is essential to solve problems involving intervals because:
Understanding cumulative probability is essential to solve problems involving intervals because:
- It adds up the probabilities of all possible outcomes up to a certain point.
- Using a cumulative probability helps to establish boundaries for specific probabilities within a distribution.
Steps in Probability Calculation
To calculate probability in the context of the standard normal distribution, we follow a straightforward approach:
In our case:
- First, identify the Z-scores involved and find their respective cumulative probabilities from the standard normal distribution table or via a calculator.
- Subtract the smaller cumulative probability (related to the lower bound Z-score) from the larger one (related to the upper bound Z-score).
In our case:
- The cumulative probability associated with 0 is 0.5 (as it aligns with the mean of the distribution).
- For , the cumulative probability is approximately 0.1093.
Effective Statistics Problem Solving
Statistics problems often involve working through a series of methodical steps. Each step relies on a thorough understanding of the concepts and tools available:
Remember, practice and familiarity with these tools and concepts will improve your problem-solving skills over time. Whether it's a classroom exercise or real-world data analysis, the principles remain the same. With clear steps and comprehension, like in our exercise, finding the probability for a range , can become both straightforward and rewarding.
- Start by clearly understanding the problem and identifying what needs to be computed, often involving Z-scores and their corresponding probabilities.
- Use tools like standardized normal distribution tables or statistical calculators to access cumulative probabilities.
- Follow through defined calculations, ensuring each component is accurate and logical.
Remember, practice and familiarity with these tools and concepts will improve your problem-solving skills over time. Whether it's a classroom exercise or real-world data analysis, the principles remain the same. With clear steps and comprehension, like in our exercise, finding the probability for a range , can become both straightforward and rewarding.
