Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 6: Problem 39
Find the probabilities for each, using the standard normal distribution. $$P(z<1.42) $$
Short Answer
Expert verified
\( P(z<1.42) \approx 0.9222 \).
Step by step solution
01
Understanding the standard normal distribution
The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is often noted as a bell curve and is used to find probabilities of specific values of a normal distribution by converting them to the standard normal form, denoted as \( z \).
02
Interpreting the probability notation
The notation \( P(z<1.42) \) indicates that we need to find the probability that the standard normal variable \( z \) is less than 1.42. This can be found using the cumulative distribution function (CDF) for the standard normal distribution.
03
Using the Z-table or Calculator
Use a Z-table or a statistical calculator to find \( P(z<1.42) \). The Z-table lists the cumulative probability of \( z \) values. Locate 1.42 on the Z-table, which typically involves finding the row for 1.4 and the column for 0.02.
04
Reading the Z-table
In the Z-table, find the intersection of the row and column corresponding to 1.4 and 0.02. This value represents \( P(z<1.42) \). According to standard Z-tables, this is approximately 0.9222.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Probability
Probability is the measure of the likelihood that a particular event will occur.
It is a fundamental concept in statistics and mathematics critically involved in decision-making processes and predicting outcomes. In the context of the standard normal distribution, probability helps us determine the likelihood that a random variable falls within a certain range.
It is a fundamental concept in statistics and mathematics critically involved in decision-making processes and predicting outcomes. In the context of the standard normal distribution, probability helps us determine the likelihood that a random variable falls within a certain range.
- It ranges from 0 to 1, where 0 indicates impossibility and 1 indicates certainty.
- Probabilities for a given event can be expressed in fractions, decimals, or percentages.
- In the standard normal distribution, probabilities can be found by calculating the area under the curve for a particular range of values.
What is the Cumulative Distribution Function?
The cumulative distribution function (CDF) is a key concept in the realm of probabilities for the standard normal distribution.
It provides a way to calculate the probability that a random variable will take a value less than or equal to a specific value, such as 1.42 in our example. The CDF is thus a running total of probability up to a certain point.
It provides a way to calculate the probability that a random variable will take a value less than or equal to a specific value, such as 1.42 in our example. The CDF is thus a running total of probability up to a certain point.
- The CDF for a standard normal distribution increases from 0 to 1 as the variable moves from negative infinity to positive infinity.
- With respect to the standard normal curve (bell curve), the CDF accumulates the area under the curve from left to right.
- In standard normal distributions, the CDF can be found using mathematical tools such as Z-tables or statistical software.
Using the Z-table
A Z-table is an essential tool utilized in statistics to find the probability of specific events when dealing with standard normal distributions.
It contains a list of cumulative probabilities of a standard normal distribution's Z-scores.
It contains a list of cumulative probabilities of a standard normal distribution's Z-scores.
- The Z-table shows the probability that a variable is less than a specific Z-score, such as 1.42 in this case.
- To use it, identify the row representing the Z-score up to one decimal place, and then locate the column for the precise hundredths value.
- For a Z-score of 1.42, find 1.4 in the row and the column for .02.
Explaining the Z-score
The Z-score is a crucial statistical measure used to describe a value's relation to the mean of a group of values.
In the standard normal distribution, it is a normalized score showing how many standard deviations a data point is from the mean (which is 0 in the standard normal distribution).
In the standard normal distribution, it is a normalized score showing how many standard deviations a data point is from the mean (which is 0 in the standard normal distribution).
- A Z-score can be positive, indicating the score is above the mean, or negative, showing it is below the mean.
- Calculating a Z-score provides a way to easily compare values from different data sets or distributions.
- It is calculated using the formula: \[Z = \frac{(X - \mu)}{\sigma}\]where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
