Normal distributions apply to many situations in real life, and you can take an average and a standard deviation to solve real world problems. Instead of working with a theoretical Z point, you will focus on a real world X point to solve. Let’s look at one example below.

Francesca is a cyclist. She wants to know the probability that she will finish the race in under 5 minutes. The racing times for this particular race are normally distributed, with an average finish time of 4 minutes and a standard deviation of 1.3 minutes.

To work on any word problem, it is important to first pull out the information you need to solve.

1.  Find out what the problem is asking!

In this case, the problem is wondering: What is the probability that Francesca will finish the race in under 5 minutes?

2. Find the necessary numbers

Population mean μ = 4

Population standard deviation σ = 1.3

Value of focus X = 5

3. Sketch the problem

We know we are looking for the probability less than 5. We also know that the mean is at the center of a normal distribution. Therefore, we can sketch the blue area to represent the probability we need to find.

4. Solve

There are several ways to approach it, with different technologies.

Now that you know what we are looking for, how do you solve? There are a few ways!

Using the formula to convert X to Z:

This formula will tell you the Z point in the Normal Distribution. Basically it turns our real world x-value into a Z point.

This is not the answer, YET! This number is used to find the area under the curve to the left of this point. You can use the “Normal Distribution Curve Document” as reference or technology!

You can utilize the normal distribution function for real world problems in Excel “Norm....” to find area under the curve in the same way as Statcrunch.

To find a point greater than X, or to the right of X, you must subtract from 100% or 1. Here, you can see the function to solve for the probability under the curve/area under the curve in Excel. Be sure to enter your specific real world X value.

You are able to enter the REAL numbers you identified in a word problem: x, mean,and standard deviation.

For a full list of statistics symbols, click here: Statistics & Probability Symbols

1. Select Stat, then Calculators, then Normal

When using the normal distribution calculator, you can find the area to the left and right of a point, in this case the X point

• Less than (≤) indicates to the left
• Greater than (≥) indicates to the right

Since we now have a mean and standard deviation, we will enter that, too! When you enter the point you want to consider, you will find the approximate area to the left or right of that point, also called the probability under the normal curve.

Example One

Taco-eating contests and Clarence’s taco eating times have been found to be random normal distributions. Clarence wants to enter the contest, but is unsure if the taco-eating skills are enough to win. In researching the contest, Clarence finds that the record for 50 tacos eaten in the contest is 4.5 minutes. Clarence eats 50 tacos in an average of 5 minutes, with a standard deviation of 1.1 tacos. What is the probability that Clarence eats 50 tacos in less than 4.5 minutes in order to win the contest?

Remember your Steps:

1. What is the problem asking?
2. Identify the given numbers
3. Sketch the curve
4. Solve the problem.

Check the solutions page, to be sure you understand the concept!

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