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Math 201 Applied Statistics

- Day 7
- 1.3 Normal Distributions

- The distribution of Salaries is Skewed to the

Right. So Mean is larger than Median. (Few

players are paid in millions while most are not) - 2. Five-number Summary
- Min, Q1, Med, Q3, Max
- 101 103 109 115 126 126 129 137 137 140 148 152

154 154 165 165 178 200 - Modified Boxplot
- IQR154 -12628
- 1.5(IQR)1.52842
- Q31.5(IQR)15442196 ? any value larger than

196 is an outlier - Q1-1.5(IQR)126-4284 ? any value smaller than 84

is an outlier

Last time we discussed

- Standard deviation (s)
- How it works
- Properties
- How the summary stats change when you do a

linear transformation

General Strategy

- Graph the data
- Look for overall pattern and deviations from this

pattern - Add numerical measures center and spread
- ?One more step
- If the pattern is regular describe it by a smooth

curve

1.3 Density curves

- Density curve
- The mathematical model for the distribution.
- The total area under the curve1

Relative freq

Ex 1.78 page 84

(No Transcript)

Mean of a Density curve

Mean is the Balance point

(No Transcript)

1.3 Normal Distributions

- Symmetric
- Bell shaped
- Unimodal

For density curves Mean is denoted by m Std dev

is denoted by s

Locating std dev s

s

s

(No Transcript)

The 68-95-99.7 Rule

- Example
- The distribution of heights of young adult women

is approximately normal with mean 64.5 inches and

standard deviation 2.5 inches. - Notation
- Height N(m, s) or Height N(64.5, 2.5)

Height N(64.5, 2.5)

m64.5 s2.5

68

Heights

2.5

2.5

One standard deviation 2.5 inches

The 68-95-99.7 Rule

- Approx. 68 of the observations fall within one

standard deviation from m

Height N(64.5, 2.5)

m64.5 s2.5

95

69.5

Heights

59.5

5

5

2 standard deviations 2 2.5 5 inches

The 68-95-99.7 Rule

- Approx. 68 of the observations fall within one

standard deviation from m - Approx. 95 of the observations fall within two

standard deviation from m - Approx. 99.7 of the observations fall within

three standard deviation from m

Height N(64.5, 2.5)

m64.5 s2.5

99.7

72

57

7.5

7.5

3 standard deviations 3 2.5 7.5 inches

Example

- The amount spent monthly on groceries has a

normal distribution with an average of 300 a

month with a standard deviation of 50. - What percentage of Americans spend between

200-400 a month? - What percentage of Americans spend less than 250

a month? - How much do the middle 99.7 of Americans spend?
- What percentage of Americans spend less than 275

a month?

Standardizing and Z-score

- Z-score tells how many standard deviations the

value falls away from the mean - Z0 ? ?
- Z positive ? ?
- Z negative ? ?

Example

- Jennie scored 600 on the verbal part of the SAT.

Her friend Gerald took the ACT and scored 21 on

the verbal part. - SAT scores are normally distributed with a mean

of 500 and standard deviation of 100. - ACT scores are normally distributed with mean 18

and standard deviation 6. - Compare Geralds and Jennies scores.

Standard Normal Distribution

Standardize

Normal Distribution

Shaded area in first graph is the same as the

shaded area in the second graph

GroceriesN(300,50)

What percentage of Americans spend less than

315.50?

Normal Distribution

Groceries

300

315.50

?

?

50

What is the shaded area?

.6217 or 62.17

.6217

0.3

.01

Homework

- 1.79, 1.83, 1.87

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