Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 6: Problem 55
\(X \sim N(-3,4)\) Find the probability that \(x\) is between one and four.
Short Answer
Expert verified
The probability that \(x\) is between 1 and 4 is approximately 0.1186.
Step by step solution
01
Define the Problem
We need to find the probability that a random variable, which follows a normal distribution with mean mu = -3 and standard deviation sigma = 4, takes on a value between 1 and 4.
02
Standardize the Values
To calculate the probability of a normal distribution, we convert the values to a standard normal distribution (Z-distribution). This involves calculating the Z-scores for the values 1 and 4 using the formula: \[ Z = \frac{X - \mu}{\sigma} \] For X = 1: \[ Z_1 = \frac{1 - (-3)}{4} = \frac{4}{4} = 1 \] For X = 4: \[ Z_2 = \frac{4 - (-3)}{4} = \frac{7}{4} = 1.75 \]
03
Use the Standard Normal Distribution Table
Now that we have the Z-scores, determine the probability of Z being less than those scores by using the standard normal distribution table (or a calculator with this functionality).
The probability that Z < 1 is approximately 0.8413 and that Z < 1.75 is approximately 0.9599.
04
Calculate the Probability P(1 < X < 4)
Now find the probability that X is between 1 and 4 by subtracting the lower tail from the upper tail. \[ P(1 < X < 4) = P(Z < 1.75) - P(Z < 1) \approx 0.9599 - 0.8413 = 0.1186 \]
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.
Z-scores
When working with normal distributions, a Z-score is a statistical measurement that describes a data point's relationship to the mean of a group of points. If you have an observation and you want to know how far away it is from the average, Z-scores are your go-to tool.
Here鈥檚 how it works:
Here鈥檚 how it works:
- A Z-score tells you how many standard deviations an element is from the mean.
- The Z-score formula is: \[ Z = \frac{X - \mu}{\sigma} \]where:
- \(X\) is the value of the data point,
- \(\mu\) is the mean,
- \(\sigma\) is the standard deviation.
- A Z-score of 0 indicates the value is exactly on the mean. Positive Z-scores indicate values above the mean, while negative Z-scores indicate values below it.
- In our case, calculating the Z-scores for X = 1 (\(Z_1 = 1\)) and X = 4 (\(Z_2 = 1.75\)), helps us translate individual scores into a common framework for comparison.
Standard Normal Distribution Table
The Standard Normal Distribution Table, often called the Z-table, is a tool used to find probabilities related to the standard normal distribution. It's a calculated table that lists the cumulative probability of a standard normal variable being below a particular Z-score.
Using the Z-table:
Using the Z-table:
- This table allows us to determine the likelihood of a variable falling below a certain level on the bell curve.
For instance, when you have a Z-score, the Z-table can help fetch the probability or the area under the curve up to that score. - For a Z-score of 1, the table gives a cumulative probability of approximately 0.8413, which implies there's an 84.13% chance the random variable is less than Z = 1.
- Similarly, a Z-score of 1.75 results in a cumulative probability of 0.9599, or 95.99% chance the variable is less than this score.
Probability Calculation
Once you have your Z-scores and the corresponding probabilities from the Z-table, calculating the probability over a range becomes straightforward.
Steps include:
Steps include:
- Identify your Z-scores for the considered range. For example, one Z-score for the lower limit (Z = 1) and one for the upper limit (Z = 1.75).
- Use the probability values from the Z-table relevant to your Z-scores.
From earlier, we have P(Z < 1) = 0.8413 and P(Z < 1.75) = 0.9599. - Subtract the probability of the lower limit from that of the upper limit: \[ P(1 < X < 4) = P(Z < 1.75) - P(Z < 1) \approx 0.9599 - 0.8413 = 0.1186 \]
Standard Deviation and Mean
Understanding standard deviation and mean is crucial for interpreting normal distributions effectively.
Mean (\(\mu\)) and standard deviation (\(\sigma\)) fundamentals:
Mean (\(\mu\)) and standard deviation (\(\sigma\)) fundamentals:
- The mean is the average of all the data points, serving as a central point on the distribution.
- Standard deviation measures how dispersed the data points are from the mean:
- A smaller standard deviation indicates data points are tightly clustered around the mean.
- A larger standard deviation means they are spread out over a wider range of values.
- In our problem, the given mean, \(\mu\), is -3, and the standard deviation, \(\sigma\), is 4. This suggests most data will hover around -3, but it can reasonably spread out 4 units in either direction.
- It is the combination of these two metrics that helps define the shape and spread of a distribution, giving context to our probability and Z-score calculations.
