Section 7.2: Applications of the Normal Distribution

1. 7.1 Properties of the Normal Distribution
2. 7.2 Applications of the Normal Distribution
3. 7.3 Assessing Normality

For a quick overview of this section, watch this short video summary:

Finding Areas Using a Table

Once we have the general idea of the Normal Distribution, the next step is to learn how to find areas under the curve. We'll learn two different ways - using a table and using technology.

Since every normally distributed random variable has a slightly different distribution shape, the only way to find areas using a table is to standardize the variable - transform our variable so it has a mean of 0 and a standard deviation of 1. How do we do that? Use the z-score!

As we noted in Section 7.1, if the random variable X has a mean μ and standard deviation σ, then transforming X using the z-score creates a random variable with mean 0 and standard deviation 1! With that in mind, we just need to learn how to find areas under the standard normal curve, which can then be applied to any normally distributed random variable.

Finding Area under the Standard Normal Curve to the Left

Before we look a few examples, we need to first see how the table works. Before we start the section, you need a copy of the table. You can download a printable copy of this table, or use the table in the back of a textbook. It should look something like this:

It's pretty overwhelming at first, but if you look at the picture at the top (take a minute and check it out), you can see that it is indicating the area to the left. That's the key - the values in the middle represent areas to the left of the corresponding z-value. To determine which z-value it's referring to, we look to the left to get the first two digits and above to the columns to get the hundredths value. (Z-values with more accuracy need to be rounded to the hundredths in order to use this table.)

Say we're looking for the area left of -2.84. To do that, we'd start on the -2.8 row and go across until we get to the 0.04 column. (See picture.)

From the picture, we can see that the area left of -2.84 is 0.0023.

Finding Areas Using StatCrunch

Click on Stat > Calculators > Normal Enter the mean, standard deviation, x, and the direction of the inequality. Then press Compute. The image below shows P(Z < 1.23). Note that for Z, the mean is 0 and the standard deviation is 1.

Let's try some examples.

Example 1

Finding Area under the Standard Normal Curve to the Right

To find areas to the right, we need to remember the complement rule. Take a minute and look back at the rule from Section 5.2.

Since we know the entire area is 1,

(Area to the right of z₀) = 1 - (Area to the left of z₀)

Example 2

An alternative idea is to use the symmetric property of the normal curve. Instead of looking to the right of Z=2.68 in Example 2 above, we could have looked at the area left of -2.68. Because the curve is symmetric, those areas are the same.

Finding Area under the Standard Normal Curve Between Two Values

To find the area between two values, we think of it in two pieces. Suppose we want to find the area between Z = -2.43 and Z = 1.81.

What we do instead, is find the area left of 1.81, and then subtract the area left of -2.43. Like this:

So the area between -2.43 and 1.81 = 0.9649 - 0.0075 = 0.9574

Example 3

Finding Areas Under a Normal Curve Using the Table

1. Draw a sketch of the normal curve and shade the desired area.
2. Calculate the corresponding Z-scores.
3. Find the corresponding area under the standard normal curve.

If you remember, this is exactly what we saw happening in the Area of a Normal Distribution demonstration. Follow the link and explore again the relationship between the area under the standard normal curve and a non-standard normal curve.

Finding Areas Under a Normal Curve Using StatCrunch

Even though there's no "standard" in the title here, the directions are actually exactly the same as those from above!

Click on Stat > Calculators > Normal Enter the mean*, standard deviation*, X, and the direction of the inequality. Press Compute. * Note that for Z, the mean is 0 and the standard deviation is 1.

Now we finally get to the real reason we study the normal distribution. We want to be able to answer questions about variables that are normally distributed. Questions like..

• What proportion of individuals are geniuses?
• Is a systolic blood pressure of 110 unusual?
• What percentage of a particular brand of light bulb emits between 300 and 400 lumens?
• What is the 90th percentile for the weights of 1-year-old boys?

All of these questions can be answered using the normal distribution!

Example 4

Example 5

Why don't you try a couple?

Example 6

Example 7

Finding Values

The next type of question comes from the other direction. Instead of giving values and asking for the probability, we'll now be looking at problems where the probability is known, but the values are not. Questions like:

• What is the 90th percentile for the weights of 1-year-old boys?
• What IQ score is below 80% of all IQ scores?
• What weight does a 1-year-old boy need to be so all but 5% of 1-year-old boys weight less than he does?

As with the previous types of problems, we'll learn how to do this using both the table and technology. Make sure you know both methods - they're both used in many fields of study!

Finding Z-Scores Using the Table

The idea here is that the values in the table represent area to the left, so if we're asked to find the value with an area of 0.02 to the left, we look for 0.02 on the inside of the table and find the corresponding Z-score.

Since we don't have an area of exactly 0.02, we have to think a bit. We have two choices: (1) take the closest area, or (2) average the two values if it's equidistant from the two areas.

In this case, it's almost equidistant, so we'll take the average and say that the Z-score corresponding to this area is the average of -2.05 and -2.06, so -2.055.

Finding Normally Distributed Values Using StatCrunch

Click on Stat > Calculators > Normal Enter the mean*, standard deviation*, the direction of the inequality, and the probability (leave X blank). Press Compute. * Note that for Z, the mean is 0 and the standard deviation is 1.

Let's try a few!

Example 8

So we've talked about how to find a z-score given an area. If you remember, the technology instructions didn't specify that the distribution needed to be the standard normal - we actually find values in any normal distribution that correspond to a given area/probability using those same techniques.

Example 9

Example 10

Example 11

Finding z_α

The notation z_α ("z-alpha") is the Z-score with an area of α to the right.

The concept of z_αis used extensively throughout the remainder of the course, so it's an important one to be comfortable with. The applications won't be immediately obvious, but the essence is that we'll be looking for events that are unlikely - and so have a very small probability in the "tail".

Let's try some examples.

Example 12

 

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