Lecture 5: Limit Laws and Calculate Limits II
1
Absolute Values
Recall The absolute value of x, denoted by |x|, is
Example 1:
1.Show that lim |x| = 0
x0
2. Does lim
x0
|x|
exist? Why?
x
1
| x2 1 |
x2 x 1
3. lim
Theorem
If g(x) = f (x) when x 6= a, then
li
1
STAT 2004
Hamdy F. F. Mahmoud
Statistics Department
Virginia Tech
Fall 2014
Part II
Observational and Experimental Study
Ways to Collect Data
There are five ways to collect data:
Well-Designed A study in which the researcher imposes conditions on the
Ex
Note: Dividing the totals by the sample size, will give
us the marginal probabilities
Grade
Male
Female
Totals
A
122
65
0
41
17
2
163/247=0.66
B
C
Total
187/247=0.76 60/247=0.24
82/247=0.33
2/247=0.01
247/247=1
Probability of C
Probability of Male
p(male)
Revision
Question #0
What is
? Example, what is
?
1
Question #1
In a group of 40 people, 10 are healthy and every person of the
remaining 30 has high blood pressure, a high level of cholesterol
or both. If 15 have high blood pressure and 25 have high leve
The Link Between Sleep Quantity and Academic
Performance for the College Student
Megan Lowry1, Kayla Dean2, and Keith Manders
Department of Psychology, University of Minnesota, Minneapolis, Minnesota
There have been many studies that link unhealthy sleep
Probability p
Table entry for p and C is
the critical value t with
probability p lying to its
right and probability C lying
between t and t .
t*
TABLE D t distribution critical values
Upper tail probability p
df
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
STAT 2004: Introductory Statistics
Basic Ideas
Statistics: Learning from Data
Statistics is the art and science of learning from data.
Art comes from the different creative and meaningful ways
to visualize, summarize, and analyze data
Science comes from t
Statistical inference
Test of Hypotheses
Hamdy Mahmoud
VT, Stat. Dep.
Fall 2014
The Five Steps of a Significance Test
p
1. Assumptions
Under what conditions the sample proportion, p, has
Normal distribution?
When n is large, np and nq 15 , CLT says p ha
3. Summarizing more than one
variable graphically and numerically
2. Summarizing one variable
numerically.
1. Summarizing one variable
graphically.
Summarizing Data
A Data Set in JMP
www.dreamstime.com
Visualizing one variable
Q: I collected data, then wh
1
STAT 2004
Hamdy F. F. Mahmoud
Statistics Department
Virginia Tech
Fall 2014
Observational and Experimental Study
Ways to Collect Data
There are five ways to collect data:
Well-Designed A study in which the researcher imposes conditions on the
Experiment
Lecture 6: Precise Definition of a Limit
2.4 The Precise Definition of a Limit
Inequalities Review
1. Absolute Value
|x| <
|x| >
|x a| <
Example 1. Consider the function y = 2x1 near x = 4. We know lim (2x1) = 7.
x4
Now lets see how close to 4 does x h
Lecture 8: Discontinuity and Continuity
1
Continuity
Recall If P(x) is a polynomial, then
lim P (x) = P (a)
xa
But this property is not true for every function. So the question is for which functions,
this property holds?
Definition:
A function f is conti
Lecture 4: Limit Laws and Calculate Limits
1
Calculating limits using the limit laws
Theorem 1 - Limit Laws
Suppose k is a constant, lim f (x) exists, and lim g(x) exists. For 15, write the
xc
xc
formula according to the verbal statement
1. Sum Rule: The
Spring 2016 STAT 4004 Homework 2
Student Name: Jonathan Costa
1. Suppose that a, b are both logical, we have learned in class that the TRUE/FALSE output of
the expression a|b depends on the TRUE/FALSE values of a and b. We also learned that under
some cir
Spring 2016 STAT 4004 Homework 1
Student Name: Jonathan Costa
1. Give R assignment statements that set the variable z to the digit in the second decimal place of
x.
Answer: 2
2. Give R expressions that return the following matrices and vectors: (b) (1, 2,
STAT 2004: Introductory of Statistics
Hamdy F. F. Mahmoud
Virginia Tech
Stat. Dep
Fall 2014
Gaming industry sets odds for lotteries based
on probability theory
Card Games (gambling)
Sporting Events (horse racing)
Weather Forecast
Suppose a Family has 4 ch
Stat 3615
Simple Linear Regression
Hamdy F. F. Mahmoud
Virginia Tech
Statistics department
Fall 2014
Regression Analysis
Regression Models
One of the most popular ideas in statistics is the idea
of a statistical model.
Statistical Model: An approximation
STAT 2004: Introductory Statistics
Sampling Methods
Hamdy Mahmoud
Virginia Tech
Statistics Department
Fall 2014
LEARNING OBJECTIVES
Develop an understanding about different sampling methods
Distinguish between random & non random sampling
Discuss the r
Summarizing one variable
numerically
Practice problem
-5
-8
The following represents sample data. Show
that the sample mean of the data is equal to
1, and the sample standard deviation of the
sample data is equal to 5.117.
-3
-2
0
1
2
2
3
4
10
8
Summariz
Guidelines for Test 2
I.
Study all the lecture notes that cover:
1. Summarizing two variables graphically and numerically
2. Probability
3. Discrete probability distributions
II.
Solve the practice problems that I uploaded on scholar. I also uploaded the
STAT 2004: Introductory Statistics
Recitation Leader: Yi Liu
Recitation 11
Question 1
Assume the population we are interested in is STAT 2004 students. Below is the Normal
probability plot, the histogram and summary statistics for Exam 2 scores, of STAT 2
Recitation activity #7 (15 points )
Oct.13-16, 2014
STAT 2004: Introductory Statistics
Your name: -
Your ID -
Your recitation leader name: -Meng Zhao-Answer all questions completely before turning in at the end of the recitation.
Question #1
You have a th
Recitation activity #8 (15 points )
Oct.20-23, 2014
STAT 2004: Introductory Statistics
Your name: -
Your ID -
Your recitation leader name: -MENG ZHAO-Answer all questions completely before turning in at the end of the recitation.
HINT: Use Vinn Diagrams t
STAT 2004: Introductory Statistics
Your name: - -5-Q.L.:Z.ZZ9_1/_§ _ _ Your ID _ _
Your recitation leader name; . _, _ _ L
Answer 3" Questions completely before turning in at the end of the recitation.
Mica#1; You toss a coin 1 time and roll a die 1 time.
Recitation #2 (20 points + 5 extra credit) September 81 1, 2014
M
STAT 2004: Introductory Statistics
Soiul'i'onr
YOUI name: - Your ID - -
Your recitation leader name: . -
Answer all questions completely before turning in at the end of the recitation.
1. F
activity #10 (15 points) Nov.3-6. 2014
STAT 2004: Statistics
Your m: mma-m Your nmo-v-o-v-Ovv-m-«n-
So/uTIOA/J
Your leader name: «mum-mm».-
Amcr all questions completeiy before turning in at the end of the recitation.
A life insmance salesman sells on th
Recitation activity #12 (15 points) Dec.l-4. 20H
_._-
STAT 2004: Introductory Statistics
Your name: you; {D . -
" "'S'EZL'EEEDE
Your recitation leader name: . -
Answer all questions completely before turning in at the end of the recitation.
Question #1: T