Despite the complexity of this function, there are only two parameters that completely deter- mine the shape of the distribution: the mean, \(\mu\text{,}\) and the standard deviation, \(\sigma\text{.}\) Let’s take a closer look at how they impact the distribution. (Don’t worry---we won’t be using this formula to compute probabilities. We have other tools!)
How does changing the standard deviation (\(\sigma\)) and the mean (\(\mu\)) change the curve? Figure7.1.3.Graph of Normal Distribution powered by Desmos
Random variables around the mean, where the curve is the tallest have a higher probability of occurring than values toward the end of the distribution.
The total area under the curve is \(1\text{.}\) Since the distribution is symmetric, the area to the left of the mean equals \(0.5\) as does the area to the right of the mean
Subsection7.1.3Examples of Normal Probability Distributions:
There are many examples of random variables that are normally distributed in many different real-world settings! Here are a few examples of random variables that are normally distributed or that could be in certain situations:
Recall that the \(z\)-score of a value, \(x\text{,}\) describes the number of standard deviations that the value \(x\) is from the mean of its distribution:
A coffee shop wants to understand the distribution of amounts that their customers are spending per order. They find that the average amount a customer spends per order is \(\$ 7.50\text{,}\) and the standard deviation is \(\$1.25\text{.}\) If we assume that the amounts spent per order are normally distributed, and Rey comes in and spends \(\$10.00\text{,}\) what is the \(z\)-score for Rey’s order?
When the original random variable, \(x\text{,}\) follows a normal distribution, \(z\)-scores also follow a normal distribution with \(\mu=0\text{,}\)\(\sigma=1\text{.}\) It is called the standard normal distribution.
Subsection7.1.5Calculating Probabilities for Normal Distributions:
The standard normal distribution allows us to calculate probabilities for any normal distribution since we can standardize it with \(z\)-scores. The tables at this link (Standard Normal Table) provide you with the cumulative area to the LEFT of the \(z\)-score.
Subsection7.1.6Calculating Normal Distribution Values in Excel:
We already saw an Excel command that will allow us to find a probability associated with a standard normal distribution. There is also an Excel command that allows us to find a probability associated with a general normal distribution, and in this case, we will need to specify the mean and standard deviation of the distribution. (These first two commands both apply to situations where the goal is to compute a probability associated with a data value.)
\(NORM.DIST(x,\text{mean},\text{standard deviation},\text{cumulative})\text{:}\) applies to problems involving a normal distribution with a specified mean and standard deviation
We’re going to do a matching activity in class to practice working with these formulas to calculate probabilities for normal distributions. (The next few exercises include online versions of part of this activity.)
Exercise7.1.14.Normal Distribution Practice Activity.
(a)
A random variable follows the normal probability distribution with a mean of 135 and a standard deviation of 22. What is the probability that a randomly selected value is more than 40?
A random variable follows the normal probability distribution with a mean of 135 and a standard deviation of 22. What is the probability that a randomly selected value is less than 90?
A random variable follows the normal probability distribution with a mean of 135 and a standard deviation of 22. What is the probability that a randomly selected value is between 120 and 180?
Subsection7.1.7Finding \(Z\) or \(X\) if we have information about a probability:
What if we want to know the specific \(X\) that satisfies a given probability? Then we can work backward using the tables (Standard Normal Table) and use the \(z\)-score formula in reverse. If the exact probability cannot be found in the table, then we can use the closest values.
Suppose the retirement age of professional athletes in a certain league is normally distributed with mean \(32.27\) years and standard deviation \(3.62\) years. Find the following:
Find \(k\) such that the probability that a randomly selected athlete’s retirement age is between \(32.27-k\) years and \(32.27+k\) years is \(42\%\text{.}\)
