1

Stat 1510: Statistical Thinking and Concepts

Density Curves and Normal Distribution

2

Topics

 Density Curves  Normal Distributions  The 68-95-99.7 Rule  The Standard Normal Distribution  Finding Normal Proportions  Using the Standard Normal Table  Finding a Value When Given a Proportion

Objectives

3

 Define and describe density curves  Measure position using percentiles  Measure position using z-scores  Describe Normal distributions  Describe and apply the 68-95-99.7 Rule  Describe the standard Normal distribution  Perform Normal calculations

Density Curves

4

In previous classes, we developed a kit of graphical and numerical tools for describing distributions. Now, we’ll add one more step to the strategy.

1.

2.

3.

4.

Exploring Quantitative Data

Always plot your data: make a graph.

Look for the overall pattern (shape, center, and spread) and for striking departures such as outliers.

Calculate a numerical summary to briefly describe center and spread.

Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve.

5

Histogram

Increase sample size, reduce the class width, then we can approximate the histogram by a smooth curve

6

Density Curves

Example:

Here is a histogram of vocabulary scores of 947 seventh graders.

The smooth curve drawn over the histogram is a

mathematical model

distribution.

for the

Density Curves

7

The areas of the shaded bars in this histogram represent the proportion of scores in the observed data that are less than or equal to 6.0. This proportion is equal to 0.303.

Now the area under the smooth curve to the left of 6.0 is shaded. If the scale is adjusted so the total area under the curve is exactly 1, then this curve is called a

density curve.

The proportion of the area to the left of 6.0 is now equal to 0.293.

8

Density Curves

A

density curve

is a curve that: • is always on or above the horizontal axis • has an area of exactly 1 underneath it A density curve describes the overall pattern of a distribution. The area under the curve and above any range of values on the horizontal axis is the proportion of all observations that fall in that range.

Density Curves

9

 Our measures of center and spread apply to density curves as well as to actual sets of observations.

Distinguishing the Median and Mean of a Density Curve

• The

median

of a density curve is the equal-areas point, the point that divides the area under the curve in half.

• The

mean

of a density curve is the balance point, at which the curve would balance if made of solid material.

• The median and the mean are the same for a symmetric density curve. They both lie at the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail.

9

10

Density Curves

 The mean and standard deviation computed from

x s

, respectively.

 The mean and standard deviation of the actual distribution represented by the density curve are denoted by

µ

( “ mu ” ) and  ( “ sigma ” ), respectively.

Normal Distributions

11

 One particularly important class of density curves are the Normal curves, which describe Normal distributions.

  All Normal curves are symmetric, single-peaked, and bell-shaped A Specific Normal curve is described by giving its mean

µ

deviation

σ

.

and standard

12

Normal Distributions

A

Normal distribution

is described by a Normal density curve. Any particular Normal distribution is completely specified by two numbers: its mean

µ

and standard deviation

σ

. • The mean of a Normal distribution is the center of the symmetric

Normal curve.

• The standard deviation is the distance from the center to the change-of-curvature points on either side.

• We abbreviate the Normal distribution with mean

µ

deviation

σ

as

N

(

µ,σ

).

and standard

Question

13

Data sets consisting of physical measurements (heights, weights, lengths of bones, and so on) for adults of the same species and sex tend to follow a similar pattern. The pattern is that most individuals are clumped around the average, with numbers decreasing the farther values are from the average in either direction. Describe what shape a histogram (or density curve) of such measurements would have?

14

The 68-95-99.7 Rule

The 68-95-99.7 Rule

In the Normal distribution with mean

µ

and standard deviation

σ

: • Approximately

68%

of the observations fall within

σ

of

µ.

• Approximately

95%

of the observations fall within 2

σ

of

µ.

• Approximately

99.7%

of the observations fall within 3

σ

of

µ.

N(0,1)

15

68-95-99.7 Rule for any Normal Curve

µ



68%

µ

+ 

µ µ

-2 

µ

-3 

99.7%

µ µ

+3 

95%

µ µ

+2 

Normal Distributions

16

 The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for 7 th -grade students in Gary, Indiana, is close to Normal. Suppose the distribution is N(6.84, 1.55).

 Sketch the Normal density curve for this distribution.

 What percent of ITBS vocabulary scores are less than 3.74?

 What percent of the scores are between 5.29 and 9.94?

17

Health and Nutrition Examination Study of 1976-1980

 Heights of adult men, aged 18-24  mean: 70.0 inches  standard deviation: 2.8 inches  heights follow a normal distribution, so we have that heights of men are N(70, 2.8).

18

Health and Nutrition Examination Study of 1976-1980

 68-95-99.7 Rule for men’s heights  68% are between 67.2 and 72.8 inches [ µ   = 70.0  2.8 ]  95% are between 64.4 and 75.6 inches [ µ  2  = 70.0  2(2.8) = 70.0  5.6 ]  99.7% are between 61.6 and 78.4 inches [ µ  3  = 70.0  3(2.8) = 70.0  8.4 ]

19

Health and Nutrition Examination Study of 1976-1980

 What proportion of men are less than 68 inches tall?

?

68 70 (height values)

How many standard deviations is 68 from 70?

The Standard Normal Distribution

20

 All Normal distributions are the same if we measure in units of size

σ

from the mean

µ

as center.

The

standard Normal distribution

is the Normal distribution with mean 0 and standard deviation 1.

If a variable

x

has any Normal distribution

N

(

µ,σ

) with mean

µ

and standard deviation

σ

, then the standardized variable

z



x

 -

μ

has the standard Normal distribution,

N

(0,1).

Key 20 | 3 means 203 pounds Stems = 10

’

s Leaves = 1

’

s

The Standard Normal Table

21

 Because all Normal distributions are the same when we standardize, we can find areas under any Normal curve from a single table.

The Standard Normal Table Table A

is a table of areas under the standard Normal curve. The table entry for each value

z

is the area under the curve to the left of

z

.

Suppose we want to find the proportion of observations from the standard Normal distribution that are less than 0.81. We can use Table A:

Z

0.7

0.8

0.9

.00

.7580

.7881

.8159

.01

.7611

.7910

.8186

.02

.7642

.7939

.8212

P(z < 0.81) = .7910

Normal Calculations

22

Find the proportion of observations from the standard Normal distribution that are between -1.25 and 0.81.

Can you find the same proportion using a different approach?

1 – (0.1056+0.2090) = 1 – 0.3146

=

0.6854

23

Revisiting Earlier Example - Health and Nutrition Examination Study of 1976-1980

 How many standard deviations is 68 from 70?



standardized score

= (observed value minus mean) / [ = (68 70) / 2.8 = 0.71

] (std dev)  The value 68 is 0.71 standard deviations below the mean 70.

24

Health and Nutrition Examination Study of 1976-1980

 What proportion of men are less than 68 inches tall?

?

68 70 (height values) -0.71 0 (standardized values)

25

Health and Nutrition Examination Study of 1976-1980

 What proportion of men are less than 68 inches tall?

.2389

68 70 (height values) -0.71 0 (standardized values)

26

Normal Calculations

How to Solve Problems Involving Normal Distributions State:

Express the problem in terms of the observed variable

x

.

Plan:

Draw a picture of the distribution and shade the area of interest under the curve.

Do:

Perform calculations.

• •

Standardize x

to restate the problem in terms of a standard Normal variable

z

.

Use Table A

and the fact that the total area under the curve is 1 to find the required area under the standard Normal curve.

Conclude:

Write your conclusion in the context of the problem.

Normal Calculations

27

 According to the Health and Nutrition Examination Study of 1976-1980, the heights (in inches) of adult men aged 18-24 are N(70, 2.8).

 How tall must a man be in the lower 10% for men aged 18 to 24?

N(70, 2.8)

.10

? 70

Normal Calculations

28

 How tall must a man be in the lower 10% for men aged 18 to 24?

.10

? 70 Look up the closest probability (closest to 0.10) in the table.

Find the corresponding

standardized score.

The value you seek is that many standard deviations from the mean.

z 1.3

-1.2

1.1

.07

.0853

.1020

.1210

.08

.0838

.1003

.1190

Z = -1.28

N(70, 2.8) .09

.0823

.0985

.1170

29

Normal Calculations

N(70, 2.8)  How tall must a man be in the lower 10% for men aged 18 to 24?

.10

Z = -1.28

? 70 We need to “unstandardize” the z-score to find the observed value (

x

):

z



x

 

x



z



x

= 70 +

z

(2.8) = 70 + [( 1.28 )  (2.8)] = 70 + ( 3.58) =

66.42

A man would have to be approximately 66.42 inches tall or less to place in the lower 10% of all men in the population.

Question

30

A company engaged in the manufacturing of special type of material. Diameter of the this component is a critical characteristic. From the previous analysis we know that diameter follows normal distribution with mean 10.05 and std deviation 0.25. The diameter specification is 10 +/ 0.4. If the company has an order of 10000 items, as a manger of the company, how many items you will schedule to produce, so that you may able to sent 10000 good items to the customer?