For A Standard Normal Distribution, Find The Approximate Value Of $P(-0.78 \leq Z \leq 1.16)$. Use The Portion Of The Standard Normal Table Below To Help Answer The Question. \[ \begin{tabular}{|c|c|} \hline Z$ & Probability
Introduction
In probability theory, the standard normal distribution, also known as the z-distribution, is a widely used distribution that has a mean of 0 and a standard deviation of 1. It is a continuous probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In this article, we will explore how to find the approximate value of the probability of a standard normal distribution within a given range.
Understanding the Standard Normal Distribution
The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. The probability density function (PDF) of the standard normal distribution is given by:
The standard normal distribution is often used as a reference distribution in statistical inference, and it is used to standardize variables by converting them to z-scores. The z-score is a measure of how many standard deviations an observation is away from the mean.
Finding the Approximate Value of a Standard Normal Distribution
To find the approximate value of a standard normal distribution within a given range, we can use the standard normal table. The standard normal table provides the probability that a standard normal random variable takes on a value less than or equal to a given z-score. We can use this table to find the probability that a standard normal random variable takes on a value between two given z-scores.
Using the Standard Normal Table
The standard normal table is a table that provides the probability that a standard normal random variable takes on a value less than or equal to a given z-score. The table is typically arranged in a grid, with the z-scores listed in the left-hand column and the corresponding probabilities listed in the top row.
To find the approximate value of a standard normal distribution within a given range, we can use the following steps:
- Find the z-scores that correspond to the lower and upper bounds of the range.
- Use the standard normal table to find the probability that a standard normal random variable takes on a value less than or equal to the lower z-score.
- Use the standard normal table to find the probability that a standard normal random variable takes on a value less than or equal to the upper z-score.
- Subtract the probability found in step 2 from the probability found in step 3 to find the approximate value of the probability that a standard normal random variable takes on a value within the given range.
Finding the Approximate Value of P(-0.78 ≤ z ≤ 1.16)
To find the approximate value of P(-0.78 ≤ z ≤ 1.16), we can use the standard normal table. The standard normal table provides the probability that a standard normal random variable takes on a value less than or equal to a given z-score.
| z | Probability |
|---|---|
| -0.78 | 0.2177 |
| 1.16 | 0.8751 |
Using the standard normal table, we can find the probability that a standard normal random variable takes on a value less than or equal to -0.78 and the probability that a standard normal random variable takes on a value less than or equal to 1.16.
Calculating the Approximate Value
To find the approximate value of P(-0.78 ≤ z ≤ 1.16), we can subtract the probability found in the standard normal table for z = -0.78 from the probability found in the standard normal table for z = 1.16.
P(-0.78 ≤ z ≤ 1.16) = P(z ≤ 1.16) - P(z ≤ -0.78) = 0.8751 - 0.2177 = 0.6574
Therefore, the approximate value of P(-0.78 ≤ z ≤ 1.16) is 0.6574.
Conclusion
In this article, we explored how to find the approximate value of a standard normal distribution within a given range. We used the standard normal table to find the probability that a standard normal random variable takes on a value less than or equal to a given z-score. We then used this table to find the approximate value of P(-0.78 ≤ z ≤ 1.16). The approximate value of P(-0.78 ≤ z ≤ 1.16) is 0.6574.
References
- Kendall, M. G., & Stuart, A. (1977). The advanced theory of statistics. Macmillan.
- Johnson, N. L., & Kotz, S. (1970). Distributions in statistics: Continuous univariate distributions. Wiley.
- Evans, M., Hastings, N., & Peacock, B. (2000). Statistical distributions. Wiley.
Further Reading
- Standard Normal Distribution Table
- Probability Density Function (PDF)
- Z-Score
- Standard Normal Distribution
Note: The standard normal table provided is a simplified version and may not be comprehensive. For a more detailed and accurate table, please refer to a standard statistics textbook or online resources.
Introduction
In the previous article, we explored how to find the approximate value of a standard normal distribution within a given range. We used the standard normal table to find the probability that a standard normal random variable takes on a value less than or equal to a given z-score. In this article, we will answer some frequently asked questions (FAQs) about standard normal distribution.
Q: What is the standard normal distribution?
A: The standard normal distribution, also known as the z-distribution, is a widely used distribution that has a mean of 0 and a standard deviation of 1. It is a continuous probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.
Q: What is the purpose of the standard normal table?
A: The standard normal table is a table that provides the probability that a standard normal random variable takes on a value less than or equal to a given z-score. It is used to find the approximate value of a standard normal distribution within a given range.
Q: How do I use the standard normal table?
A: To use the standard normal table, you need to find the z-scores that correspond to the lower and upper bounds of the range. Then, you use the table to find the probability that a standard normal random variable takes on a value less than or equal to the lower z-score and the probability that a standard normal random variable takes on a value less than or equal to the upper z-score. Finally, you subtract the probability found in the first step from the probability found in the second step to find the approximate value of the probability that a standard normal random variable takes on a value within the given range.
Q: What is the difference between the standard normal distribution and the normal distribution?
A: The standard normal distribution and the normal distribution are both continuous probability distributions, but they have different means and standard deviations. The standard normal distribution has a mean of 0 and a standard deviation of 1, while the normal distribution has a mean of μ and a standard deviation of σ.
Q: How do I convert a normal distribution to a standard normal distribution?
A: To convert a normal distribution to a standard normal distribution, you need to standardize the variable by converting it to a z-score. The z-score is a measure of how many standard deviations an observation is away from the mean.
Q: What is the z-score?
A: The z-score is a measure of how many standard deviations an observation is away from the mean. It is calculated by subtracting the mean from the observation and dividing the result by the standard deviation.
Q: How do I use the z-score to find the approximate value of a standard normal distribution?
A: To use the z-score to find the approximate value of a standard normal distribution, you need to find the z-scores that correspond to the lower and upper bounds of the range. Then, you use the standard normal table to find the probability that a standard normal random variable takes on a value less than or equal to the lower z-score and the probability that a standard normal random variable takes on a value less than or equal to the upper z-score. Finally, you subtract the probability found in the first step from the probability found in the second step to find the approximate value of the probability that a standard normal random variable takes on a value within the given range.
Q: What are some common applications of the standard normal distribution?
A: The standard normal distribution has many applications in statistics and data analysis, including:
- Hypothesis testing: The standard normal distribution is used to test hypotheses about the mean and standard deviation of a population.
- Confidence intervals: The standard normal distribution is used to construct confidence intervals for the mean and standard deviation of a population.
- Regression analysis: The standard normal distribution is used to analyze the relationship between variables in a regression model.
- Time series analysis: The standard normal distribution is used to analyze and forecast time series data.
Conclusion
In this article, we answered some frequently asked questions (FAQs) about standard normal distribution. We discussed the purpose and use of the standard normal table, the difference between the standard normal distribution and the normal distribution, and some common applications of the standard normal distribution. We hope that this article has provided you with a better understanding of the standard normal distribution and its applications.
References
- Kendall, M. G., & Stuart, A. (1977). The advanced theory of statistics. Macmillan.
- Johnson, N. L., & Kotz, S. (1970). Distributions in statistics: Continuous univariate distributions. Wiley.
- Evans, M., Hastings, N., & Peacock, B. (2000). Statistical distributions. Wiley.
Further Reading
- Standard Normal Distribution Table
- Probability Density Function (PDF)
- Z-Score
- Standard Normal Distribution
Note: The standard normal table provided is a simplified version and may not be comprehensive. For a more detailed and accurate table, please refer to a standard statistics textbook or online resources.