# Exercise for calculating the upper cut-off point in a normal distributionInverse Normal Distribution Example

### Problem Data

It is required to calculate the limit of the random variable X given the **right tail probability** taking into consideration the following data:

- Mean (μ)
- 60
- Standard Deviation (σ)
- 15
- Probability (Area)
- 0.1
- Description
**Exercise for calculating the upper cut-off point in a normal distribution**A very large group of students obtains scores (from 0 to 100) that follow a normal distribution having a mean of 60 and a standard deviation of 15. Find the cutoff point of the top 10 percent of all students.

## Solution

Below are the detailed calculations and graphs to obtain **the limit** according to the data provided:

### Step 1:

We have the normal random variable X with mean μ and variance σ^{2}; that is, ** X ~ N(μ,σ^{2})**, which will become the standard normal random variable,

**. To do so, we will apply the following transformation:**

*Z ~ N(0,1)*We have μ = 60 and σ = 15, then:

Given the right tail boundary * b*, we have the following:

Once the expression has been transformed to a normal distribution, it is necessary to know for which value of Z this probability is obtained. Performing computer calculations, the exact value of Z to obtain a right-tailed probability of 0.1 is **1.2816**.

Using this value, we replace the result in the previous equation:

**Calculation with table**

You can also use your statistical table to solve the problem, however the result may not be exact (it will be an approximation), because the statistical table only considers Z values with two decimal places. The procedure will be as follows:

**Adapt the expression to the left tail:**Since the table we use shows the area to the left, the symmetry of the normal distribution must be used to fit it:**Search in the table matrix:**In the table results, find the number closest to 0.9 and mark it. In our table the closest number is 0.8997.- From the marked number, we locate the values of its corresponding
**row=1.2**and**column=0.08**. Finally, we add both values and obtain the result**1.28**

The right tail cut-off point where the random variable X has a probability of 0.1 is **79.224** and its approximation with statistical table is **79.2**

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