# 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 distributionA 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 σ2; that is, X ~ N(μ,σ2), which will become the standard normal random variable, Z ~ N(0,1). To do so, we will apply the following transformation:

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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