More applications of the Z-Score

NORMAL DistributionThe Empirical Rule

Normal Distribution?These density curves are symmetric, single-peaked, and bell-shaped.We capitalize Normal to remind you that these curves are special.

Normal distribution is described by giving its mean μ and its standard deviation σ

Shape of the Normal curve

The standard deviation σ controls the spread of a Normal curve

The Empirical Rule68-95-99.7 rule

In the Normal distribution with mean μ and standard deviation σ:

• Approximately 68% of the observations fall within σ of the mean μ.

• Approximately 95% of the observations fall within 2σ of μ.

• Approximately 99.7% of the observations fall within 3σ of μ.

The Normal Distribution and Empirical Rule

Example: YOUNG WOMEN’s HEIGHT

The distribution of heights of young women aged 18 to 24 is approximately Normal with mean μ = 64.5 inches and standard deviation σ = 2.5 inches.

Importance of Normal Curve

• scores on tests taken by many people (such as SAT exams and many psychological tests),

• repeated careful measurements of the same quantity, and

• characteristics of biological populations (such as yields of corn and lengths of animal pregnancies).

even though many sets of data follow a Normal distribution, many do not.

Most income distributions, for example, are skewed to the right and so are not Normal

Standard Normal distribution

Standard Normal DistributionThe standard Normal distribution is the Normal distribution N(0, 1) with mean 0 and standard deviation

Standard Normal Calculation

The Standard Normal TableTable 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.

Area to the LEFTUsing the standard Normal table

Problem: Find the proportion of observations from the standard Normal distribution that are less than 2.22.

illustrates the relationship between the value z = 2.22 and the area 0.9868.

How to use the table of values

illustrates the relationship between the value z = 2.22 and the area 0.9868.

Example 

Area to the RIGHTUsing the standard Normal table

Problem: Find the proportion of observations from the standard Normal distribution that are greater than −2.15

z = −2.15

Area = 0.0158

Area = 1-0.0158

Area = .9842

Practice

(a) z < 2.85

(b) z > 2.85

(c) z > −1.66

(d) −1.66 < z < 2.85

(a) 0.9978.

(b)0.0022.

(c) 0.9515.

(d) 0.9493.

CODY’S quiz score relative to his classmates

79 81 80 77 73 83 74 93 78 80 75 67 73

77 83 86 90 79 85 89 84 77 72 83 82 x

z = 0.99

Area = .8389

Cody’s actual score relative to the other students who took

the same test is 84%