Problem 12 Let \(Z\) be a Normal \((0,1)\) ... [FREE SOLUTION] (2024)

Short Answer

Cumulative Distribution Function

The Cumulative Distribution Function (CDF) is a key concept in statistics, especially when dealing with the normal distribution. The CDF of a random variable, like the standard normal variable \(Z\), gives the probability that \(Z\) will take a value less than or equal to some specified value.
For instance, if you want to know the probability that \(Z\) is less than or equal to \(3.06\), you look up \(\text{Φ}(3.06)\) in the CDF.
Mathematically, it is written as \[ P(Z \leq z) = \text{Φ}(z) \] where \( P(Z \leq z) \) is the probability and \( z \) is the specified value.
Understanding the CDF is crucial because it directly relates to finding probabilities in the standard normal distribution.

Probability

Probability is the measure of the likelihood that an event will occur. In our context, the event is \(Z\) being in an interval.
When dealing with normal distributions, probabilities often refer to areas under the curve of the normal distribution graph. For example, the probability that \(Z \leq 3.06\) can be visualized as the area of the graph from \(-∞\) to \(3.06\).
This probability is found using the CDF of the normal distribution. Simply put, if \( \text{Φ}(3.06) = 0.9989 \), it means there is a 99.89% chance that \(Z\)'s value will be less than or equal to \(3.06\).

Standard Normal Table

The Standard Normal Table, also known as the Z-table, is a very useful tool for finding probabilities and cumulative probabilities for the standard normal distribution.
This table lists values of \( \text{Φ}(z) \) for different \(z\) values, which are the CDF values of those \(z\) values.
For example, if you look up \(z = 3.06\) in the standard normal table, you'll find that \( \text{Φ}(3.06) = 0.9989 \).
You can use the table to quickly find probabilities without needing to calculate integrals yourself. Nowadays, you can also use online versions of the Z-table, or built-in functions in calculative tools.

Statistical Software

Statistical software refers to various applications and online tools that help perform statistical analyses quickly and accurately.
These tools are very helpful when dealing with normal distribution and other statistical calculations. Popular examples include R, Python (with libraries like SciPy or NumPy), and online calculators.
To find \( \text{Φ}(3.06) \), you can type relevant commands in R or Python, or simply enter it in an online normal distribution calculator to instantly get the value 0.9989.
These tools save time and reduce the risk of manual calculation errors.