For A Standard Normal Distribution, Find The Approximate Value Of P ( Z \textless 1.20 P(z \ \textless \ 1.20 P ( Z \textless 1.20 ]. Use The Portion Of The Standard Normal Table Below To Help Answer The Question. \[ \begin{tabular}{|c|c|} \hline Z$ & Probability
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. In this article, we will explore how to find the approximate value of the probability of a standard normal distribution for a given z-score.
What is a Standard Normal Distribution?
A standard normal distribution is a continuous probability distribution that is symmetric about the mean, which is 0. The distribution is characterized by a bell-shaped curve, with the majority of the data points concentrated around the mean and tapering off gradually towards the extremes. The standard deviation of the distribution is 1, which means that about 68% of the data points fall within one standard deviation of the mean, about 95% fall within two standard deviations, and about 99.7% fall within three standard deviations.
Using the Standard Normal Table
To find the approximate value of the probability of a standard normal distribution for a given z-score, we can use the standard normal table. The table 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 organized in a way that the z-scores are listed in the left-hand column, and the corresponding probabilities are listed in the top row.
Finding the Approximate Value of P(z < 1.20)
To find the approximate value of P(z < 1.20), we can use the standard normal table. We need to locate the row corresponding to z = 1.20 and read off the probability from the table.
| z | Probability |
|---|---|
| 0.00 | 0.5000 |
| 0.01 | 0.5040 |
| 0.02 | 0.5080 |
| 0.03 | 0.5119 |
| 0.04 | 0.5159 |
| 0.05 | 0.5199 |
| 0.06 | 0.5239 |
| 0.07 | 0.5279 |
| 0.08 | 0.5319 |
| 0.09 | 0.5359 |
| 0.10 | 0.5399 |
| 0.11 | 0.5439 |
| 0.12 | 0.5479 |
| 0.13 | 0.5519 |
| 0.14 | 0.5559 |
| 0.15 | 0.5599 |
| 0.16 | 0.5639 |
| 0.17 | 0.5679 |
| 0.18 | 0.5719 |
| 0.19 | 0.5759 |
| 0.20 | 0.5799 |
| 0.21 | 0.5839 |
| 0.22 | 0.5879 |
| 0.23 | 0.5919 |
| 0.24 | 0.5959 |
| 0.25 | 0.5999 |
| 0.26 | 0.6039 |
| 0.27 | 0.6079 |
| 0.28 | 0.6119 |
| 0.29 | 0.6159 |
| 0.30 | 0.6199 |
| 0.31 | 0.6239 |
| 0.32 | 0.6279 |
| 0.33 | 0.6319 |
| 0.34 | 0.6359 |
| 0.35 | 0.6399 |
| 0.36 | 0.6439 |
| 0.37 | 0.6479 |
| 0.38 | 0.6519 |
| 0.39 | 0.6559 |
| 0.40 | 0.6599 |
| 0.41 | 0.6639 |
| 0.42 | 0.6679 |
| 0.43 | 0.6719 |
| 0.44 | 0.6759 |
| 0.45 | 0.6799 |
| 0.46 | 0.6839 |
| 0.47 | 0.6879 |
| 0.48 | 0.6919 |
| 0.49 | 0.6959 |
| 0.50 | 0.6999 |
| 0.51 | 0.7039 |
| 0.52 | 0.7079 |
| 0.53 | 0.7119 |
| 0.54 | 0.7159 |
| 0.55 | 0.7199 |
| 0.56 | 0.7239 |
| 0.57 | 0.7279 |
| 0.58 | 0.7319 |
| 0.59 | 0.7359 |
| 0.60 | 0.7399 |
| 0.61 | 0.7439 |
| 0.62 | 0.7479 |
| 0.63 | 0.7519 |
| 0.64 | 0.7559 |
| 0.65 | 0.7599 |
| 0.66 | 0.7639 |
| 0.67 | 0.7679 |
| 0.68 | 0.7719 |
| 0.69 | 0.7759 |
| 0.70 | 0.7799 |
| 0.71 | 0.7839 |
| 0.72 | 0.7879 |
| 0.73 | 0.7919 |
| 0.74 | 0.7959 |
| 0.75 | 0.7999 |
| 0.76 | 0.8039 |
| 0.77 | 0.8079 |
| 0.78 | 0.8119 |
| 0.79 | 0.8159 |
| 0.80 | 0.8199 |
| 0.81 | 0.8239 |
| 0.82 | 0.8279 |
| 0.83 | 0.8319 |
| 0.84 | 0.8359 |
| 0.85 | 0.8399 |
| 0.86 | 0.8439 |
| 0.87 | 0.8479 |
| 0.88 | 0.8519 |
| 0.89 | 0.8559 |
| 0.90 | 0.8599 |
| 0.91 | 0.8639 |
| 0.92 | 0.8679 |
| 0.93 | 0.8719 |
| 0.94 | 0.8759 |
| 0.95 | 0.8799 |
| 0.96 | 0.8839 |
| 0.97 | 0.8879 |
| 0.98 | 0.8919 |
| 0.99 | 0.8959 |
| 1.00 | 0.9000 |
| 1.01 | 0.9039 |
| 1.02 | 0.9079 |
| 1.03 | 0.9119 |
| 1.04 | 0.9159 |
| 1.05 | 0.9199 |
| 1.06 | 0.9239 |
| 1.07 | 0.9279 |
| 1.08 | 0.9319 |
| 1.09 | 0.9359 |
| 1.10 | 0.9399 |
| 1.11 | 0.9439 |
| 1.12 | 0.9479 |
| 1.13 | 0.9519 |
| 1.14 | 0.9559 |
| 1.15 | 0.9599 |
| 1.16 | 0.9639 |
| 1.17 | 0.9679 |
| 1.18 | 0.9719 |
| 1.19 | 0.9759 |
| 1.20 | 0.9799 |
From the table, we can see that the probability of a standard normal random variable taking on a value less than or equal to 1.20 is approximately 0.9799.
Conclusion
In this article, we have explored how to find the approximate value of the probability of a standard normal distribution for a given z-score. We have used the standard normal table to find the probability of a standard normal random variable taking on a value less than or equal to 1.20, which is approximately 0.9799. This value can be used in a variety of applications, including hypothesis testing and confidence intervals.
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.
- Johnson, R. A., & Bhattacharyya, G. K. (2014). Statistics: Principles and methods. John Wiley &
Frequently Asked Questions (FAQs) about Standard Normal Distribution ====================================================================
In this article, we will answer some frequently asked questions about 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. 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.
Q: What is the difference between a standard normal distribution and a normal distribution?
A: The main difference between a standard normal distribution and a normal distribution is the mean and standard deviation. A standard normal distribution has a mean of 0 and a standard deviation of 1, while a normal distribution can have any mean and standard deviation.
Q: How is the standard normal distribution used in real-life applications?
A: The standard normal distribution is used in a variety of real-life applications, including:
- Hypothesis testing: The standard normal distribution is used to determine the probability of observing a certain value or range of values in a sample.
- Confidence intervals: The standard normal distribution is used to construct confidence intervals for population parameters.
- Regression analysis: The standard normal distribution is used to model the relationship between a dependent variable and one or more independent variables.
- Time series analysis: The standard normal distribution is used to model the behavior of time series data.
Q: How do I use the standard normal table to find probabilities?
A: To use the standard normal table to find probabilities, you need to locate the row corresponding to the z-score you are interested in and read off the probability from the table. The table provides the probability that a standard normal random variable takes on a value less than or equal to a given z-score.
Q: What is the relationship between the standard normal distribution and the normal distribution?
A: The standard normal distribution is a special case of the normal distribution. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Any normal distribution can be transformed into a standard normal distribution by subtracting the mean and dividing by the standard deviation.
Q: How do I calculate the z-score for a given value?
A: To calculate the z-score for a given value, you need to subtract the mean and divide by the standard deviation. The formula for calculating the z-score is:
z = (X - μ) / σ
where X is the value, μ is the mean, and σ is the standard deviation.
Q: What is the significance of the standard normal distribution in statistics?
A: The standard normal distribution is a fundamental concept in statistics and is used to model a wide range of phenomena in various fields. It is used in hypothesis testing, confidence intervals, regression analysis, and time series analysis. The standard normal distribution is also used to calculate probabilities and z-scores.
Q: How do I use the standard normal distribution to model real-life phenomena?
A: To use the standard normal distribution to model real-life phenomena, you need to:
- Identify the mean and standard deviation of the phenomenon you are interested in.
- Use the standard normal table to find the probability of observing a certain value or range of values.
- Use the z-score formula to calculate the z-score for a given value.
- Use the standard normal distribution to model the behavior of the phenomenon.
Conclusion
In this article, we have answered some frequently asked questions about standard normal distribution. We have discussed the definition, properties, and applications of the standard normal distribution. We have also provided examples of how to use the standard normal table to find probabilities and how to calculate z-scores.