Answer
The 11th percentile is 48.64
SOLUTION
Problem statement
We are told to find the 11th percentile of a random variable X that is normally distributed with a mean of 56 and a standard deviation of 6.
Method
- The percentile represents the probability that variable X will take a particular value; only that the probability has been multiplied by 100% to make it a percentile.
- This implies that:
- The normal distribution gives us a probability density curve with the mean at its center, the values of variable X on the x-axis while the corresponding Z-score is shown on the y-axis.
- Thus, the probability of 0.11 tells us that the area under the probability density curve is 0.11 and we are looking for the value of X that corresponds to this area of 0.11.
This is illustrated below:
- Thus, to find the value of X that corresponds to the 11th percentile (or Area under the curve of 0.11), we follow these steps:
1. Find the left-hand side Z-score that corresponds to the probability of 0.11.
2. Apply the Z-score formula to find the value of X. The Z-score formula is given by:
Let us apply these steps and find the value of X.
Implementation
Step 1:
Find the left-hand side Z-score:
We shall find this Z-score using Z-score tables as shown below:
- From the above figure, we can observe that the probability 0.11 lies between a Z-score of -1.22 and -1.23.
- We can decide the exact number using interpolation or by using an online Z-score calculator.
- Using an online Z-score calculator, we have that Z(0.11) = - 1.227.
Step 2: Apply the Z-score formula:
Final Answer
The 11th percentile is 48.64