# Lesson 3 Y B Normal? Solidify Understanding

## Learning Focus

Understand a scale used to compare normal distributions.

Sketch distribution curves and use tables to find population percentages.

How can we compare normal distributions that have different means and standard deviations?

## Open Up the Math: Launch, Explore, Discuss

As a college admissions officer, you get to evaluate hundreds of applications from students who want to attend your school. Many of them have good grades, have participated in school activities, have done service within their communities, and have all kinds of other attributes that would make them great candidates for attending the college you represent. One part of the application that is considered carefully is the applicant’s score on the college entrance examination. At the college you work for, some students have taken the ACT and some students have taken the SAT.

You need to make a final decision on two applicants. They are both wonderful students with the very same G.P.A. and class rankings. It all comes down to their test scores. Student A took the ACT and received a score of

### 1.

Based only on their test scores, which student would you choose and why?

This analysis is starting to make you hungry, so you call your friend in the Statistics Department at the university and ask her to go to lunch with you. During lunch, you tell her of your dilemma. The conversation goes something like this:

**You:** I’m not sure that I’m making the right decision about which of two students to admit to the university. Their entrance exam scores seem like they’re in about the same part of the distribution, but I don’t know which one is better. It’s like trying to figure out which bag of fruit weighs more when one is measured in kilograms and one is measured in pounds. They might look like about the same amount, but you can’t tell the exact difference unless you put them on the same scale or convert them to the same units.

**Statistician:** Actually, there is a way to make comparisons on two different normal distributions that is like converting the scores to the same unit. The scale is called the “standard normal distribution.” Since it was invented to make it easy to think about a normal distribution, they set it up so that the mean is

Here’s what your statistician friend drew on her napkin to show you the standard normal distribution:

**You:** Well, that looks just like the way I always think of normal distributions.

**Statistician:** Yes, it’s pretty simple. When we use this scale, we give things a

Try it: Let’s say you had two imaginary test takers, Student C and Student D. Student C’s

### 2.

What percent of the test takers scored below Student C?

### 3.

What percent of the test takers scored below Student D?

### 4.

What percent of the test-takers scored between Student C and Student D?

### 5.

Student C and Student D have a friend, Student E, who scored

Pause and Reflect

**You:** That’s very cool, but the two scores I’m working with are not given as

**Statistician:** Sure. The scale wouldn’t be so amazing if you couldn’t use it for any normal distribution. There’s a little formula for transforming a data point from any normal distribution to a standard normal distribution:

### 6.

So, if you have an ACT score of

### 7.

Let’s use the formula to figure it out:

**You**: That’s great. I’m going back to the office to decide which student is admitted.

### 8.

Compare the scores of Student A and Student B. Explain which student has the highest mathematics test score and why.

## Ready for More?

Find the original ACT score for a student whose

## Takeaways

The standard normal distribution:

A

The value from a

The

## Vocabulary

- standard normal distribution
- symbols for sample statistics and corresponding population parameters
- z-score
**Bold**terms are new in this lesson.

## Lesson Summary

In this lesson, we learned about the standard normal distribution and the

Two hundred students at the newly opened Center High School were surveyed regarding their choice for the new school’s colors. Their preferences are recorded in the table.

Seniors | Juniors | Totals | |
---|---|---|---|

Red and White | |||

Purple and Gold | |||

Orange and Black | |||

Totals |

### 1.

If the sample is a true representation of a projected enrollment of

### 2.

If red and white is eliminated as a choice and the survey is taken again with the same

### 3.

Without using technology, sketch the graph of the polynomial function with the given characteristics. Write the equation in factored form.

Degree:

Roots:

Equation: