# Lesson 3Y 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 in mathematics. Student B took the SAT and received a score of in mathematics. Since you are an expert in college entrance exams, you know that both tests are designed to be normally distributed. A perfect ACT score is . The ACT mathematics section has a mean of and standard deviation of . (Source: National Center for Education Statistics 2010) A perfect score on the SAT math section is . The SAT mathematics section has a mean of and a standard deviation of . (Source: www.collegeboard.com 2010 Profile).

### 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 and the standard deviation 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 -score. A -score of means that it’s standard deviation above the mean. A -score of means that it is between and standard deviations below the mean. Easy-peasy. What’s even better is that when we have a -score, there are tables that will show the area under the curve to the left of that score, which gives you the percentage of the population. For a test score like the ACT or SAT, it shows the percentage of the population (or sample) that is below that score. I’ve got a -score table right here in my purse. See, the -score is , so approximately of the population scored less.

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

### 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 . Find the percent of test-takers who scored above them without using a table or technology. Explain your strategy.

Pause and Reflect

You: That’s very cool, but the two scores I’m working with are not given as -scores. Is there some way that I can transform values from some normal distribution like the scores on the ACT or SAT to -scores?

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 . The mean score on the ACT is and the standard deviation is . What would you estimate the -score to be?

### 7.

Let’s use the formula to figure it out: -score = . How was your estimate? Explain why this value is reasonable.

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.

Find the original ACT score for a student whose -score is .

## Takeaways

The standard normal distribution:

A -score describes:

The value from a -score table describes:

The -score formula:

-score =

## Lesson Summary

In this lesson, we learned about the standard normal distribution and the -score, which is a method for putting all normal distributions on the same scale so they can be compared. Once a -score is computed for a given value, a -score table is used to find the population percentage to the left of the value.

## Retrieval

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 , how many students will be happy with orange and black for the school colors?

### 2.

If red and white is eliminated as a choice and the survey is taken again with the same , do you think orange and black will still be the top choice? Justify your answer.

### 3.

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

• Degree:

• Roots:

Equation: