CHAPTER 2: THE NORMAL DISTRIBUTIONS

RECALL SECTION 2.1 In section 2.1 density curves were introduced:

A density curve is an idealized mathematical model that describes the overall pattern of a distribution The area under the curve represents a proportion

of all observations and is therefore equal to 1 More specifically, we looked at the basic layout of

the normal curve Symmetric, bell shaped, unimodal distribution in

which the mean is equal to the median The 68-95-99.7 rule was used to help describe

proportions within 3 standard deviations above and below the mean

The notation used for a normal distribution is in the form N(mean, standard deviation) or 2( , )N

SECTION 2.2: STANDARD NORMAL CALCULATIONS

The standard normal distributionHas a mean of 0 and standard deviation of 1.

N(0,1)

Taking any normal distribution and converting it to have a mean of 0 and StDev of 1 is called standardizing.

A standardized value is called a z-score.

3

STANDARDIZING AND Z-SCORES. A z-score tells us how many standard

deviations an observation falls away from the mean, and in which direction. Observations larger than the mean have

positive z-scores, while observations smaller than the mean have negative z-scores.

To standardize a value, subtract the mean of the distribution from the observation and then divide by the standard deviation.

4

xz

EXAMPLE 1 – STANDARDIZING WOMEN’S HEIGHTS

The heights of young women are approximately normal with inches and inches. What is the standardized height for a woman that is

68 inches tall?

This means that a woman who is 68 inches tall has a standardized height that is 1.4 deviations above the mean

5

64.5 2.5

68 64.51.4

2.5z

1.40

xz

EXAMPLE 1 – STANDARDIZING WOMEN’S HEIGHTS

The heights of young women are approximately normal with inches and inches. What is the standardized height for a woman that is

5 feet tall?

A woman who is 5 feet tall has a standardized height that is 1.8 deviations below the mean

6

60 64.51.8

2.5z

64.5 2.5

-1.80

NORMAL DISTRIBUTION CALCULATIONS:

7

Recall that the area under a density curve represents a proportion of observations.

Since all normal distributions are the same when standardized:We can use a single table to find any area

under the curveTable A – first page of textbook.

The standard normal table will always give the area to the LEFT of the z-score

EXAMPLE 2 – USING THE STANDARD NORMAL TABLE

Using the z-scores found in example 1, where women’s heights can be described with a distribution of N(64.5, 2.5), determine the proportion of women who are:a) Shorter than 68 inches:

8

68 64.51.4

2.5z

( 68) .9192P x

The proportion of young women that are shorter than 68 inches is 91.92%

EXAMPLE 2 – USING THE STANDARD NORMAL TABLE Using the z-scores found in example 1, where

women’s heights can be described with a distribution of N(64.5, 2.5), determine the proportion of women who are:b) Taller than 5ft:

9

The proportion of young women that are taller than 5 ft. is 96.41%

60 64.51.8

2.5z

Since the table give the area to the left of the z-score, we need to subtract that proportion from 1 to get the area to the right of the z-score.

( 60) .0359P x

( 60) 1 .0359 .9641P x

COMPLETE RESPONSE TO A NORMAL DISTRIBUTION QUESTION1.State the problem in terms of the observed variable x. Draw a picture of the distribution and shade the area of interest under the curve.

2.Standardize x to restate the problem in terms of a z-score. On the picture label the Z-score.

3.Find the required area under the standard normal curve by using table A, and the fact that the total area under the curve is 1.

4.Write your conclusion in the context of the problem.

10

EXAMPLE 3- IS CHOLESTEROL A PROBLEM WITH YOUNG BOYS? The level of cholesterol in the blood is important because high

cholesterol levels may increase risk of heart disease. The distribution of blood cholesterol levels in a large population of people of the same age and sex are roughly normal. For 14-year-old boys, the mean is milligrams of cholesterol per deciliter of blood (mg/dl) and the standard deviation is Levels above 240 mg/dl may require medical attention. What percent of 14-year-old boys have more than 240 mg/dl of cholesterol? State the problem:

Call the level of cholesterol in the blood x. The variable x has the N(170, 30) distribution. We want the proportion of boys with cholesterol level x >

240

Standardize x and draw a picture:

11

170 330 mg/dl.

240 1702.33

30z

( 240)P x

EXAMPLE 3- IS CHOLESTEROL A PROBLEM WITH YOUNG BOYS?

Use the table:From the table we see that the proportion

of observations less than 2.33 (or 240mg/dl) is .9901

Thus, to the right of 2.33 is 1 - .9901 = .0099This is about .01 or 1%

Write your conclusion in the context of the problem: Only about 1% of boys have high cholesterol

12

EXAMPLE 4 – WORKING WITH AN INTERVAL

What percent of 14-year-old boys have blood pressure between 164 and 220 mg/dl?State the problem:

Standardize and draw a picture:

13

(164 220)P x

164 170.2

30z

220 1701.67

30z

Use the table:The area below 1.67 is .9525The are below -.2 is .4207To find the area between subtract the

area below -.2 from the area below 1.67.9525 - .4207 = .5318

State your conclusion in context:About 53.18% of boys have cholesterol

levels between 164 and 220 mg/dl14

EXAMPLE 4 – WORKING WITH AN INTERVAL

SECTION 2.2 DAY 1

Homework: p.95-114 #’s 19, 21, 23a & b, 28, 31a&b, 32, & 45

15