Download Basic Statistical Concepts

Basic Statistical Concepts
population–the collection of all items of interest to a researcher.
sample–a subset of the population which we gather information
on. A common sample is SRS (simple random sample).
descriptive statistics–summarize information contained in a
sample.
statistical inference–generalize from the sample to the
population.
statistic–a numerical summary of a sample. A statistic is random.
Its distribution is called sampling distribution.
Descriptive Statistics
frequency distribution and histogram: for summarizing quantitative
data.
Pn
yi
sample mean ȳ = i=1
n
sample median: the midpoint
of the ordered data.
Pn
(y
−ȳ
)2
i
2
sample variance: s = i=1n−1
.
√
sample standard deviation: s = s 2 .
Random Variables
random variable is a mapping from every possible outcome of an
experiment to real numbers. notation: X, Y, Z,...
Example: Ask a student whether she/he works part time or not.
S= {Yes, No}.
X=1 if Yes, X=0 if No.
Y = number of car accidents in a week.
Flip a coin three times. Let Z=number of heads in three flips.
{TTT } → Z = 0
{HTT , THT , TTH} → Z = 1
{HHT , HTH, THH} → Z = 2
{HHH} → Z = 3.
W = the weight of a randomly selected athlete.
Discrete Random Variables
A discrete random variable takes finite or countably infinite
values.
The probability mass function distribution (pmf) of Z is
z
P(z)
——————
0
1/8
1
3/8
2
3/8
3
1/8
Note that
1.) 0P≤ P(x) ≤ 1
2.)
P(x) = 1.
The Mean of X
The mean or expected
value of a random variable X is
P
E (X ) = µX = xP(x).
What is the mean of Z ?
The variance of P
X is
2
Var (X ) = σX = (x − µ)2 P(x) = E (X 2 ) − µ2 .
Exercise
A Computer shop builds shipments of parts it receives from various
suppliers. Let X be the number of defective hard drives per
shipments. It is assumed X has the following distribution.
x
P(x)
——————–
0
0.55
1
0.15
2
0.10
3
0.10
4
0.05
5
0.05
Find P(X ≥ 2), E (X ).