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Magazine Chapter 6: Problem 11
Given that \(z\) is a standard normal random variable, compute the following
probabilities.
a. \(\quad P(z \leq-1.0)\)
b. \(\quad P(z \geq-1)\)
c. \(\quad P(z \geq-1.5)\)
d. \(P(z \geq-2.5)\)
e. \(\quad P(-3
Short Answer
Expert verified
a. 0.1587; b. 0.8413; c. 0.9332; d. 0.9938; e. 0.4987.
Step by step solution
01
Understanding the Standard Normal Distribution
A standard normal distribution has a mean of 0 and a standard deviation of 1. Probability calculations involving the standard normal distribution involve using Z-scores and looking up probabilities in a standard normal table or computing using software.
02
Homework Requirement Analysis
We have to find the probabilities for different conditions:
- For a: Find probability that z is less than or equal to -1.
- For b, c, and d: Find probabilities that z is greater than or equal to each of -1, -1.5, -2.5.
- For e: Find probability that z is between -3 and 0.
03
Calculate P(z 鈮� -1.0)
Referring to the standard normal distribution table or using a calculator, find the probability of Z being less than or equal to -1.0. - From the standard normal table: \[ P(z \leq -1.0) \approx 0.1587 \]
04
Calculate P(z 鈮� -1)
To find P(z 鈮� -1), you need the probability of the opposite condition and subtract from 1:- Since \( P(z \leq -1) \approx 0.1587 \), - \[ P(z \geq -1) = 1 - P(z \leq -1) \approx 1 - 0.1587 = 0.8413 \]
05
Calculate P(z 鈮� -1.5)
Find the probability of Z being greater than or equal to -1.5:- From the standard normal table: - \( P(z \leq -1.5) \approx 0.0668 \)- \[ P(z \geq -1.5) = 1 - P(z \leq -1.5) = 1 - 0.0668 = 0.9332 \]
06
Calculate P(z 鈮� -2.5)
Use the Z-table for probability less than -2.5 and subtract from 1:- \( P(z \leq -2.5) \approx 0.0062 \)- \[ P(z \geq -2.5) = 1 - P(z \leq -2.5) = 1 - 0.0062 = 0.9938 \]
07
Calculate P(-3 < z 鈮� 0)
Find the probability of Z being between -3 and 0:- \( P(z \leq 0) \approx 0.5 \) since it's the mean for the standard normal distribution.- \( P(z \leq -3) \approx 0.0013 \)- \[ P(-3 < z \leq 0) = P(z \leq 0) - P(z \leq -3) = 0.5 - 0.0013 = 0.4987 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Probability Calculations
Probability calculations are essential in understanding how likely an event is to occur within a set range of values. In the context of the Standard Normal Distribution, these calculations often involve determining the probability that a Z-score falls within a certain range.
To perform these calculations, you need to understand what a Z-score is and how it relates to the normal distribution. Consider the distribution as a "bell curve." Probabilities are the areas under this curve.
To perform these calculations, you need to understand what a Z-score is and how it relates to the normal distribution. Consider the distribution as a "bell curve." Probabilities are the areas under this curve.
- For any given Z-score, find the probability of the score existing to the left or right of this point.
- Use tables or computational tools to assist in these calculations.
Z-scores
Z-scores are a fundamental concept in statistics, particularly when working with the standard normal distribution. A Z-score shows how many standard deviations an element is from the mean. The formula for calculating a Z-score is:
\[ Z = \frac{(X - \mu)}{\sigma} \]
Where:
For the standard normal distribution, both \(\mu\) and \(\sigma\) are 0 and 1 respectively, simplifying many calculations. The Z-score allows for comparison across different distributions by standardizing scores to a common scale. When you see a Z-score, think of it as a measure of unusualness. Higher or lower scores suggest rarity or dominance in data.
\[ Z = \frac{(X - \mu)}{\sigma} \]
Where:
- \(X\) is the value of the element,
- \(\mu\) is the mean,
- \(\sigma\) is the standard deviation.
For the standard normal distribution, both \(\mu\) and \(\sigma\) are 0 and 1 respectively, simplifying many calculations. The Z-score allows for comparison across different distributions by standardizing scores to a common scale. When you see a Z-score, think of it as a measure of unusualness. Higher or lower scores suggest rarity or dominance in data.
Normal Distribution Table
The Normal Distribution Table, often referred to as the Z-table, aids in probability calculations involving Z-scores. It lists the cumulative probability of a score up to and including a Z-score. This tool is invaluable for quickly determining the probability of a standard normal variable.
The table generally shows probabilities only for positive Z-scores, covering \(z > 0\). However, due to symmetry, probabilities for negative values can be found as well:
Using a Z-table involves locating the row for your Z-score and the column for its decimal place, intersecting them gives the cumulative probability.
The table generally shows probabilities only for positive Z-scores, covering \(z > 0\). However, due to symmetry, probabilities for negative values can be found as well:
- For any \(z\), the opposite side of the mean's probability is \(1 - P(Z \leq z)\).
- It provides a straightforward mechanism to find probabilities and reduce computational tasks to table lookup.
Using a Z-table involves locating the row for your Z-score and the column for its decimal place, intersecting them gives the cumulative probability.
Statistical Analysis
Statistical Analysis involves the collection and interpretation of data to discover patterns and trends. In probability and statistics, the standard normal distribution is frequently used to simplify and unify analysis.
By using Z-scores, you bring a wide range of statistical inquiries under a single framework. This is crucial for making decisions based on data, as it allows analysts to:
Understanding these principles of statistical analysis empowers you to critique past performances and forecast future outcomes efficiently. It helps in turning raw data into insights and drives informed decision-making.
By using Z-scores, you bring a wide range of statistical inquiries under a single framework. This is crucial for making decisions based on data, as it allows analysts to:
- relate individual observations to the overall population easily,
- determine probabilities quickly,
- make predictions or test hypotheses within a given confidence level.
Understanding these principles of statistical analysis empowers you to critique past performances and forecast future outcomes efficiently. It helps in turning raw data into insights and drives informed decision-making.
