Find The Following Probabilities Based On The Standard Normal Variable $Z$.Note: You May Find It Useful To Reference The $z$ Table. Round Your Answers To 4 Decimal Places.a. $P(-0.67 \leq Z \leq -0.23$\]b. $P(0 \leq Z \leq
Introduction
The standard normal variable Z is a crucial concept in statistics, and understanding how to find probabilities with it is essential for making informed decisions in various fields. In this article, we will explore how to find probabilities based on the standard normal variable Z, using the z-table as a reference. We will also discuss the importance of rounding answers to 4 decimal places.
Understanding the Standard Normal Variable Z
The standard normal variable Z is a continuous random variable that follows a normal distribution with a mean of 0 and a standard deviation of 1. The probability density function (PDF) of the standard normal variable Z is given by:
f(z) = (1/√(2π)) * e(-z2/2)
where e is the base of the natural logarithm, and π is a mathematical constant approximately equal to 3.14159.
Using the Z-Table to Find Probabilities
The z-table is a table that lists the probabilities of the standard normal variable Z falling within a certain range. The table is typically organized in a way that the z-score is listed in the leftmost column, and the corresponding probability is listed in the top row. To find the probability of the standard normal variable Z falling within a certain range, we need to look up the z-scores in the z-table and add the corresponding probabilities.
Finding Probability a: P(-0.67 ≤ Z ≤ -0.23)
To find the probability of the standard normal variable Z falling within the range -0.67 ≤ Z ≤ -0.23, we need to look up the z-scores -0.67 and -0.23 in the z-table.
| z-score | P(Z ≤ z) |
|---|---|
| -0.67 | 0.2486 |
| -0.23 | 0.4090 |
Using the z-table, we can see that the probability of the standard normal variable Z falling below -0.67 is 0.2486, and the probability of the standard normal variable Z falling below -0.23 is 0.4090. To find the probability of the standard normal variable Z falling within the range -0.67 ≤ Z ≤ -0.23, we need to subtract the probability of the standard normal variable Z falling below -0.67 from the probability of the standard normal variable Z falling below -0.23.
P(-0.67 ≤ Z ≤ -0.23) = P(Z ≤ -0.23) - P(Z ≤ -0.67) = 0.4090 - 0.2486 = 0.1604
Therefore, the probability of the standard normal variable Z falling within the range -0.67 ≤ Z ≤ -0.23 is approximately 0.1604.
Finding Probability b: P(0 ≤ Z ≤ 1.5)
To find the probability of the standard normal variable Z falling within the range 0 ≤ Z ≤ 1.5, we need to look up the z-scores 0 and 1.5 in the z-table.
| z-score | P(Z ≤ z) |
|---|---|
| 0 | 0.5 |
| 1.5 | 0.9332 |
Using the z-table, we can see that the probability of the standard normal variable Z falling below 0 is 0.5, and the probability of the standard normal variable Z falling below 1.5 is 0.9332. To find the probability of the standard normal variable Z falling within the range 0 ≤ Z ≤ 1.5, we need to subtract the probability of the standard normal variable Z falling below 0 from the probability of the standard normal variable Z falling below 1.5.
P(0 ≤ Z ≤ 1.5) = P(Z ≤ 1.5) - P(Z ≤ 0) = 0.9332 - 0.5 = 0.4332
Therefore, the probability of the standard normal variable Z falling within the range 0 ≤ Z ≤ 1.5 is approximately 0.4332.
Conclusion
In this article, we have discussed how to find probabilities based on the standard normal variable Z using the z-table. We have also discussed the importance of rounding answers to 4 decimal places. By following the steps outlined in this article, you should be able to find probabilities with the standard normal variable Z with ease.
References
- Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
- Ross, S. M. (2014). Introduction to probability models. Academic Press.
Glossary
- Standard normal variable Z: A continuous random variable that follows a normal distribution with a mean of 0 and a standard deviation of 1.
- Z-table: A table that lists the probabilities of the standard normal variable Z falling within a certain range.
- Probability density function (PDF): A function that describes the probability of a continuous random variable falling within a certain range.
Frequently Asked Questions (FAQs) about Finding Probabilities with the Standard Normal Variable Z =============================================================================================
Q: What is the standard normal variable Z?
A: The standard normal variable Z is a continuous random variable that follows a normal distribution with a mean of 0 and a standard deviation of 1.
Q: Why is the z-table important?
A: The z-table is a table that lists the probabilities of the standard normal variable Z falling within a certain range. It is an essential tool for finding probabilities with the standard normal variable Z.
Q: How do I use the z-table to find probabilities?
A: To use the z-table to find probabilities, you need to look up the z-scores in the z-table and add the corresponding probabilities.
Q: What is the difference between P(Z ≤ z) and P(-z ≤ Z ≤ z)?
A: P(Z ≤ z) is the probability of the standard normal variable Z falling below a certain z-score, while P(-z ≤ Z ≤ z) is the probability of the standard normal variable Z falling within a certain range.
Q: How do I find the probability of the standard normal variable Z falling within a certain range?
A: To find the probability of the standard normal variable Z falling within a certain range, you need to subtract the probability of the standard normal variable Z falling below the lower z-score from the probability of the standard normal variable Z falling below the upper z-score.
Q: What is the importance of rounding answers to 4 decimal places?
A: Rounding answers to 4 decimal places is important because it ensures that the answers are accurate and precise.
Q: Can I use the z-table to find probabilities for any type of normal distribution?
A: No, the z-table is only applicable for the standard normal distribution. If you need to find probabilities for a non-standard normal distribution, you will need to use a different table or calculator.
Q: How do I find the z-score for a given probability?
A: To find the z-score for a given probability, you need to look up the probability in the z-table and find the corresponding z-score.
Q: What is the relationship between the z-score and the probability?
A: The z-score is a measure of how many standard deviations away from the mean a value is, while the probability is a measure of the likelihood of a value falling within a certain range.
Q: Can I use the z-table to find probabilities for discrete random variables?
A: No, the z-table is only applicable for continuous random variables. If you need to find probabilities for a discrete random variable, you will need to use a different table or calculator.
Q: How do I choose the correct z-table for my needs?
A: To choose the correct z-table for your needs, you need to consider the type of normal distribution you are working with and the level of precision you require.
Q: What are some common applications of the z-table?
A: The z-table has many common applications in statistics, including hypothesis testing, confidence intervals, and regression analysis.
Q: Can I use the z-table to find probabilities for non-normal distributions?
A: No, the z-table is only applicable for normal distributions. If you need to find probabilities for a non-normal distribution, you will need to use a different table or calculator.
Q: How do I interpret the results of a z-table calculation?
A: To interpret the results of a z-table calculation, you need to understand the meaning of the z-score and the probability. The z-score tells you how many standard deviations away from the mean a value is, while the probability tells you the likelihood of a value falling within a certain range.