Which Of The Following Probabilities Is The Greatest For A Standard Normal Distribution?A. P ( − 1.5 ≤ Z ≤ − 0.5 P(-1.5 \leq Z \leq -0.5 P ( − 1.5 ≤ Z ≤ − 0.5 ]B. P ( − 0.5 ≤ Z ≤ 0.5 P(-0.5 \leq Z \leq 0.5 P ( − 0.5 ≤ Z ≤ 0.5 ]C. P ( 0.5 ≤ Z ≤ 1.5 P(0.5 \leq Z \leq 1.5 P ( 0.5 ≤ Z ≤ 1.5 ]D. P ( 1.5 ≤ Z ≤ 2.5 P(1.5 \leq Z \leq 2.5 P ( 1.5 ≤ Z ≤ 2.5 ]
The standard normal distribution, also known as the z-distribution, is a type of normal distribution with a mean of 0 and a standard deviation of 1. It is a fundamental concept in statistics and is used to model a wide range of phenomena in various fields, including finance, engineering, and social sciences.
Properties of the Standard Normal Distribution
The standard normal distribution has several important properties that make it a useful tool for statistical analysis. Some of these properties include:
- Symmetry: The standard normal distribution is symmetric about the mean, which is 0.
- Bell-shaped: The standard normal distribution has a bell-shaped curve, with the majority of the data points concentrated around the mean.
- Mean: The mean of the standard normal distribution is 0.
- Standard Deviation: The standard deviation of the standard normal distribution is 1.
Understanding Probability in the Standard Normal Distribution
Probability in the standard normal distribution is calculated using the z-score formula:
z = (X - μ) / σ
where X is the value of interest, μ is the mean, and σ is the standard deviation.
Calculating Probabilities in the Standard Normal Distribution
To calculate probabilities in the standard normal distribution, we can use a standard normal distribution table or calculator. The table provides the probability that a random variable takes on a value less than or equal to a given z-score.
Comparing the Given Probabilities
Let's compare the given probabilities:
A. B. C. D.
To determine which probability is the greatest, we need to calculate each probability using a standard normal distribution table or calculator.
Calculating Probability A
Using a standard normal distribution table, we can find the probability that a random variable takes on a value less than or equal to -1.5, which is approximately 0.0668. We can also find the probability that a random variable takes on a value less than or equal to -0.5, which is approximately 0.3085. To find the probability that a random variable takes on a value between -1.5 and -0.5, we subtract the probability that a random variable takes on a value less than or equal to -0.5 from the probability that a random variable takes on a value less than or equal to -1.5:
P(-1.5 ≤ z ≤ -0.5) = P(z ≤ -1.5) - P(z ≤ -0.5) ≈ 0.0668 - 0.3085 ≈ 0.2413
Calculating Probability B
Using a standard normal distribution table, we can find the probability that a random variable takes on a value less than or equal to 0.5, which is approximately 0.6915. We can also find the probability that a random variable takes on a value less than or equal to -0.5, which is approximately 0.3085. To find the probability that a random variable takes on a value between -0.5 and 0.5, we subtract the probability that a random variable takes on a value less than or equal to -0.5 from the probability that a random variable takes on a value less than or equal to 0.5:
P(-0.5 ≤ z ≤ 0.5) = P(z ≤ 0.5) - P(z ≤ -0.5) ≈ 0.6915 - 0.3085 ≈ 0.3830
Calculating Probability C
Using a standard normal distribution table, we can find the probability that a random variable takes on a value less than or equal to 1.5, which is approximately 0.9332. We can also find the probability that a random variable takes on a value less than or equal to 0.5, which is approximately 0.6915. To find the probability that a random variable takes on a value between 0.5 and 1.5, we subtract the probability that a random variable takes on a value less than or equal to 0.5 from the probability that a random variable takes on a value less than or equal to 1.5:
P(0.5 ≤ z ≤ 1.5) = P(z ≤ 1.5) - P(z ≤ 0.5) ≈ 0.9332 - 0.6915 ≈ 0.2417
Calculating Probability D
Using a standard normal distribution table, we can find the probability that a random variable takes on a value less than or equal to 2.5, which is approximately 0.9938. We can also find the probability that a random variable takes on a value less than or equal to 1.5, which is approximately 0.9332. To find the probability that a random variable takes on a value between 1.5 and 2.5, we subtract the probability that a random variable takes on a value less than or equal to 1.5 from the probability that a random variable takes on a value less than or equal to 2.5:
P(1.5 ≤ z ≤ 2.5) = P(z ≤ 2.5) - P(z ≤ 1.5) ≈ 0.9938 - 0.9332 ≈ 0.0606
Determining the Greatest Probability
Comparing the calculated probabilities, we can see that:
- P(-1.5 ≤ z ≤ -0.5) ≈ 0.2413
- P(-0.5 ≤ z ≤ 0.5) ≈ 0.3830
- P(0.5 ≤ z ≤ 1.5) ≈ 0.2417
- P(1.5 ≤ z ≤ 2.5) ≈ 0.0606
The greatest probability is P(-0.5 ≤ z ≤ 0.5) ≈ 0.3830.
Conclusion
The standard normal distribution is a fundamental concept in statistics, and it is used to model a wide range of phenomena in various fields. Here are some frequently asked questions (FAQs) about the standard normal distribution:
Q: What is the standard normal distribution?
A: The standard normal distribution, also known as the z-distribution, is a type of normal distribution with a mean of 0 and a standard deviation of 1.
Q: What are the properties of the standard normal distribution?
A: The standard normal distribution has several important properties, including:
- Symmetry: The standard normal distribution is symmetric about the mean, which is 0.
- Bell-shaped: The standard normal distribution has a bell-shaped curve, with the majority of the data points concentrated around the mean.
- Mean: The mean of the standard normal distribution is 0.
- Standard Deviation: The standard deviation of the standard normal distribution is 1.
Q: How is the standard normal distribution used in statistics?
A: The standard normal distribution is used in statistics to:
- Model real-world phenomena: The standard normal distribution is used to model a wide range of phenomena, including the heights of people, the weights of animals, and the scores of students.
- Calculate probabilities: The standard normal distribution is used to calculate probabilities, including the probability that a random variable takes on a value less than or equal to a given z-score.
- Make inferences: The standard normal distribution is used to make inferences about a population based on a sample of data.
Q: How do I calculate probabilities in the standard normal distribution?
A: To calculate probabilities in the standard normal distribution, you can use a standard normal distribution table or calculator. The table provides the probability that a random variable takes on a value less than or equal to a given z-score.
Q: What is the z-score formula?
A: The z-score formula is:
z = (X - μ) / σ
where X is the value of interest, μ is the mean, and σ is the standard deviation.
Q: How do I determine which probability is the greatest?
A: To determine which probability is the greatest, you need to calculate each probability using a standard normal distribution table or calculator and compare the results.
Q: What is the greatest probability for a standard normal distribution?
A: The greatest probability for a standard normal distribution is P(-0.5 ≤ z ≤ 0.5) ≈ 0.3830.
Q: What are some common applications of the standard normal distribution?
A: Some common applications of the standard normal distribution include:
- Finance: The standard normal distribution is used to model stock prices, interest rates, and other financial variables.
- Engineering: The standard normal distribution is used to model the performance of machines, the strength of materials, and other engineering variables.
- Social Sciences: The standard normal distribution is used to model the behavior of people, the performance of students, other social science variables.
Q: What are some common mistakes to avoid when working with the standard normal distribution?
A: Some common mistakes to avoid when working with the standard normal distribution include:
- Not using the correct z-score formula: Make sure to use the correct z-score formula, which is z = (X - μ) / σ.
- Not using a standard normal distribution table or calculator: Make sure to use a standard normal distribution table or calculator to calculate probabilities.
- Not understanding the properties of the standard normal distribution: Make sure to understand the properties of the standard normal distribution, including its symmetry, bell-shaped curve, mean, and standard deviation.
Conclusion
In conclusion, the standard normal distribution is a fundamental concept in statistics, and it is used to model a wide range of phenomena in various fields. Understanding the properties and how to calculate probabilities is essential for statistical analysis. By following the tips and avoiding common mistakes, you can effectively use the standard normal distribution in your work.