STA1501/101/3/2011
IMPORTANT INFORMATION
Read this tutorial letter rst.
DEPARTMENT OF STATISTICS
STA1501
Descriptive Statistics and Probability
Tutorial letter 101 for 2011
SCHEME OF WORK, STUDY RESOURCES AND ASSIGNMENTS
2
CONTENTS
Page
1. A word of welco
STA1501/101/3/2013
Tutorial Letter 101/3/2013
Descriptive Statistics and Probability
STA1501
Semesters 1 & 2
Department of Statistics
IMPORTANT INFORMATION:
This tutorial letter contains important
information about your module and
includes the assignment
Probability Distributions
Probability
Distributions
Ch. 6
Discrete
Probability
Distributions
Binomial
Poisson
Continuous
Probability
Distributions
Normal
Ch. 7
Discrete Random Variables
Can only assume a countable number of values
Examples:
Roll a die twice
Let X be the number of times 4 comes up
(then X could be 0, 1, or 2 times)
Toss a coin 5 times.
Let X be the number of heads
(then X = 0, 1, 2, 3, 4, or 5
Discrete Probability Distribution
Experiment: Toss 2 Coins.
T
T
H
H
T
H
T
H
Probability Distribution
X Value
Probability
0
1/4 = 0.25
1
2/4 = 0.50
2
1/4 = 0.25
Probability
4 possible outcomes
Let X = # heads.
0.50
0.25
0
1
2
X
Five-Number Summary
Box Plot: A Graphical display of data using 5number summary:
Minimum - Q1 - Median - Q3 - Maximum
Example:
25%
Minimum
Minimum
25%
1st
1st
Quartile
Quartile
25%
Median
Median
25%
3rd
3rd
Quartile
Quartile
Maximum
Maximum
Shape of Box P
Events
Simple event
Complement of an event A (denoted A)
An outcome from a sample space with one
characteristic
e.g., A red card from a deck of cards
All outcomes that are not part of event A
e.g., All cards that are not diamonds
Joint event
Involves two
Introduction to Probability
Distributions
Random Variable
Represents a possible numerical value from
an uncertain event
Random
Variables
Ch. 6
Discrete
Random Variable
Continuous
Random Variable
Ch. 7
Collectively Exhaustive Events
Collectively exhaustive events
One of the events must occur
The set of events covers the entire sample space
example:
A = aces; B = black cards;
C = diamonds; D = hearts
Events A, B, C and D are collectively exhaustive
(but
Rule of Combinations
The number of combinations of selecting X
objects out of n objects is
n!
n Cx
X! (n X)!
where:
n! =(n)(n - 1)(n - 2) . . . (2)(1)
X! = (X)(X - 1)(X - 2) . . . (2)(1)
0! = 1
(by definition)
Discrete Random Variable
Summary Measures
Expected Value (or mean) of a discrete
distribution (Weighted Average)
N
E(X) Xi P( Xi )
i1
Example: Toss 2 coins,
X = # of heads,
compute expected value of X:
E(X) = (0 x 0.25) + (1 x 0.50) + (2 x 0.25)
= 1.0
Comparing the Population with its
Sampling Distribution
Population
N=4
21
2.236
Sample Means Distribution
n=2
X 21
X 1.58
_
P(X)
.3
P(X)
.3
.2
.2
.1
.1
0
18
20
22
24
A
B
C
D
X
0
18 19
20 21 22 23
24
_
X
Sampling Distributions
A sampling distribution is a
distribution of all of the possible
values of a statistic for a given size
sample selected from a population
Continuous Probability Distributions
A continuous random variable is a variable that
can assume any value on a continuum (can
assume an uncountable number of values)
thickness of an item
time required to complete a task
temperature of a solution
height,
Probability Distributions
Probability
Distributions
Ch. 6
Discrete
Probability
Distributions
Binomial
Poisson
.
Continuous
Probability
Distributions
Normal
Ch. 7
The Poisson Distribution
Apply the Poisson Distribution when:
You wish to count the number of times an event
occurs in a given area of opportunity
The probability that an event occurs in one area of
opportunity is the same for all areas of opportunity
Mutually Exclusive Events
Mutually exclusive events
Events that cannot occur together
example:
A = queen of diamonds; B = queen of clubs
Events A and B are mutually exclusive
Z Scores
A measure of distance from the mean (for example, a
Z-score of 2.0 means that a value is 2.0 standard
deviations from the mean)
The difference between a value and the mean, divided
by the standard deviation
A Z score above 3.0 or below -3.0 is co
Which measure of location
is the best?
Mean is generally used, unless
extreme values (outliers) exist
Then median is often used, since
the median is not sensitive to
extreme values.
Example: Median home prices may be
reported for a region less sensitive t
Shape of a Distribution
Describes how data are distributed
Measures of shape
Symmetric or skewed
Left-Skewed
Symmetric
Right-Skewed
Mean < Median
Mean = Median
Median < Mean
Measures of Variation
Variation
Range
Interquartile
Range
Variance
Standard
Deviation
Coefficient
of Variation
Measures of variation give
information on the spread
or variability of the data
values.
Same center,
different variation
Tables and Charts for
Categorical data
Categorical Data
Graphing Data
Tabulating Data
The Summary Table
Pie Charts
CD
S a vin g s
Pareto Diagram
Bar Charts
B onds
S toc k s
45
0
10
20
30
40
50
120
40
100
35
30
80
25
60
20
15
40
10
20
5
0
0
S toc k s
B ond
Review Example
Five houses on a hill by the beach
$2,000 K
House Prices:
$2,000,000
500,000
300,000
100,000
100,000
$500 K
$300 K
$100 K
$100 K
Review Example:
Summary Statistics
House Prices:
$2,000,000
500,000
300,000
100,000
100,000
Mean:
Median: middl
UNlVERSITY EXAMINATIONS UNIVERSlTElTSEKSAMENS
Sc:
U N l SA mm
STA1 501 1 476060) OctoberINovember 2015
DESCRIPTIVE STATISTICS AND PROBABILITY
Duration 2 Hours 100 Marks
EXAMINERS:
FIRST MR R SSEKUMA
SECOND. MS 3 MUCHENGETWA
Use of a non-programmable poc
UNIVERSITY EXAMINATIONS UNIVERSITEITSEKSAMENS
Sig
U N l SA 2mm.
STA1501 (430935) October/November 2014
DESCRIPTIVE STATISTICS AND PROBABILITY
DuratIon 2 Hours 100 Marks
EXAMINERS'
FIRST MR R SSEKUMA
SECOND MS 8 MUCHENGETWA
Use of a non-programmable pock
STA1501/103/3/2016
Tutorial Letter 103/3/2016
Descriptive statistics and probability
STA1501
Semesters 1 & 2
Department of Statistics
CONTENT
1.
2.
Trial Examination Paper Questions
Trial Examination Paper Solutions
BAR CODE
Learn without limits.
universi
Section 7.1
Inference for the mean of a population
The t distribution:
Change: Population s.d. sigma unknown.
The t distribution
One-sample t confidence interval
One-sample t test
Matched pairs t procedures
Robustness of t procedures
The goal is to e
4.3 Measures of relative standing and linear relationships
4.3.1 Percentile
The Pth percentile is the value for which P percent are less than that value and (100-P)% are
P
greater than that value. The location of the percentile is given by L p (n 1)
100
T