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Chapter 1

Drawing Statistical Conclusions

STAT 3022

School of Statistic, University of Minnesota

January 27, 2013

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Outline

Some BasicsSummary statistics

sample mean X=n

i=1Xi/n

sample standard deviation

ni=1

(Xi X

)2/(n 1)

median, Q1, Q3, interquartile range (IQR)

IQR = Q3 Q1

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Some Basics

Roll a 6-face dice 3 times, outcome: 1, 2, 6

Sample Mean: X=

1+2+6

3 = 3Population Mean: =?

Law of large numbers: Xn for n

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Some Basics

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Some BasicsGraphical summary

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Case Study: Observational Experiment

Question: Did a bank discriminatorily pay higher startingsalaries to men than to women? Data: Beginning salaries for 32

men, 61 women. All skilled, entry-level employees hiredbetween 1969 and 1977 Perform exploratory data analysis using

graphical and numerical summaries of the data.

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Graphical Summary

Male

Starting Salary

Frequency

3000 4000 5000 6000 7000 8000 9000

0

4

8

12

Female

Starting Salary

Frequency

3000 4000 5000 6000 7000 8000 9000

0

5

10

2

0

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O tli

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Interpreting Histograms

Relative frequency histograms allow us to visually displaygeneral characteristics of the data distribution of a particularvariable:

Central tendency - Do men tend to be paid higher than

women?Spread - What is the range of most salaries?

Symmetry - Is there a skew in either distribution? Are thereany outliers?

Histograms are used to show broad features, not exquisitedetail

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Numerical Summary

lec1_1.R lec1_2.R in-class

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Normal Distribution

bell shaped, defined by the formula 12

e (x)

2

22

two parameters: mean , variance 2 (standard deviation =

2)

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Normal Distribution

Normal distribution N(, ) is defined by

f(x) =1

2e (x)

2

22

Standard normal distribution N(0, 1)

(x) =12

e12x2

Why is standard normal distribution important

f(x) =1

x

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Normal Distribution

Why normal distribution is so important?

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Normal Distribution

What is this distribution?

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Central Limit Theorem

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> dnorm(0, mean = 0, sd = 1) # density

[1] 0.3989423

> dnorm(0, mean = 0, sd = 2)

[1] 0.1994711

>

> pnorm(1, mean = 0, sd = 1) # distribution function

[1] 0.8413447

> pnorm(1, mean = 0, sd = 1, lower.tail = FALSE)

[1] 0.1586553

>

> qnorm(0.5, mean = 2, sd = 1) # quantile function

[1] 2

> qnorm(0, mean = 2, sd = 1)

[1] -Inf

>> rnorm(5, mean = 0, sd = 1) # random generation

[1] 2.2867947 1.3311000 1.9408290 -0.5366956 1.1687528

> rnorm(5)

[1] -0.48693373 0.02950848 -1.03232990 -0.24314950 -0.42515522

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Outline

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