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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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l
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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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