The Normal Distribution
It's a function that represents the distribution of many random variables as a symmetrical bell-shaped graph,
A normal distribution is defined by two parameters: The mean (μ) and the standard deviation (σ).
To calculate probabilities with the normal distribution we use technology in the form of a statistic calculator, an online site, known tools like Excel, etc.
These tools calculate the probability of occurrence of a value less than or equal to X, P(x ≤ X) which is known as the left tail of the bell curve.
It's common for these tools to require the calculation of a so-called z-score:
[tex]z=\frac{X-\mu}{\sigma}[/tex]
We are given the distribution of the foot lengths of women with a mean of μ = 22.9 cm and a standard deviation of σ = 1.3 cm.
It's required to find the probability of the US women having a shoe size of X = 22.4 cm or smaller.
Calculate the z-score:
[tex]z=\frac{22.4-22.9}{1.3}=-0.3846[/tex]
Now we use Excel to calculate P(z ≤ -0.3846) = 0.35027
The percentage is 35.0%
