ASSIGNMENT SOLVED
4. Why is a 99% confidence interval wider than a 95% confidence interval?
Answer. A 95% Confidence Interval would be narrower than a 99% Confidence Interval. This
occurs because the as the precision of the confidence interval increases (
American University of Afghanistan (AUAF)
Department of Math and Science
Statistics -1
Summer Semester 2015
Instructor: Asadullah Jawid (M. Sc. University of Bonn)
Email: ajawid@auaf.edu.af
Office: Faculty Building 2, Office No. 01.
Chapter 1
Basic Defini
3-3 Cumulative Distribution Function
1
2
Example 3-7
3
3-4 Mean and Variance of a Discrete Random Variable
4
Example 3-9
5
6
Example 3-11
3-4 Mean and Variance of a Discrete
Random Variable
See examples 3-10, 3-11, and 3-12
3-5 Discrete Uniform Distributi
2-2 Interpretations of Probability
2-2.1 Introduction
Probability
Used to quantify likelihood or chance
Repeat an Experiment many times, assume that each trial of the experiment results
in only one outcome
Frequency ratio (or Relative frequency) = the nu
2-1 Sample Spaces and Events
2-1.1 Random Experiments
Controlled Variables
Input
Text
System
Output
Uncontrolled Variables
(noise)
Because of the uncontrollable inputs, the
same settings for the controllable inputs
do not result in identical outputs every
Conditional Probability Distributions
Two Discrete Random Variables
When two random variables are defined in a random experiment,
knowledge of one can change the probabilities that we associate
with the values of the other.
2
3
4
Two Continuous Random Var
4-5 Continuous Uniform Random Variable
Definition
4-5 Continuous Uniform Random Variable
Mean and Variance
4-5 Continuous Uniform
Random Variable
4-5 Continuous Uniform Random Variable
4-6 Normal Distribution
The Normal distribution ( also known as Gaussi
3-1 Random Variables
- Random (Stochastic) experiment is defined as that whose
outcome cannot be predicted for sure
- The occurrence of a specific outcome of a stochastic process
( or an experiment) is called a random event. We use a random
variable to ma
2-6 Independence
Definition (two events)
Also,
Example 2-29
What is P(B)?
Does P(B|A) = P(B)?
2
Example 2-30 Text
D denotes the event that a part is defective
F denotes the event that a part has a surface flaw
P(D|F) denotes the probability of D assuming
3-9 Poisson Distribution
A widely used discrete probability distribution; Consider the following conditions:
- p is very small and approaches 0
example: a 100 sided dice instead of a 6 sided dice, p = 1/100 instead of 1/6
example: a 1000 sided dice, p = 1
Continuous Random Variable:
is a random variable with an interval ( either
finite or infinite) of real numbers for its range
Examples: length, electric current, time, temperature,.
The probability distribution of a continuous random variable X is
characte
1
2
4
5
6
Example
Example
4-7 Normal Approximation to the
Binomial and Poisson Distributions
The normal distribution can be used to approximate the binomial
distribution when n is large.
For large values of n it is difficult to calculate the probability
4-8 Exponential Distribution
f(x)
x
2
4-8 Exponential Distribution
The exponential distribution is sometimes used to model the waiting
time to an event. It turns out that the exponential distribution is the
correct model for waiting times whenever the eve
Independence:Two Discrete Random Variables
Joint and marginal probability
distributions of X and Y
Conditional probability distribution
of Y given X=x
Independence:Two Continuous Random Variables
3
Example:
4
Example:
5
Covariance and Correlation
Example:
Joint Probability Distribution,
2
Example: (Jointly discrete RV),
consider
The joint probability mass function is the function f XY(x,y)=P(X=x,
Y=y), for example, fXY(129,15)= 0.12
3
4
5
6
Example:
8
5-1 Two Discrete Random Variables
5-1.2 Marginal Probab
For a finite population with N equally likely values, the probability mass function is
f(xi) = 1/N and the mean is
The greater the amount of variability in the data, the larger in absolute magnitude of the deviations
Sample variance
-> 1/(N-1)
Population
1
- Often in practice we are interested in drawing valid conclusions about a large group
of individuals or objects.
- Instead of examining the entire group, called the population, which may be
difficult, we may examine only a small part of this population
Chapter 12
Inference on Categorical Data
12.1 Goodness of Fit Test
1. These procedures are for testing whether sample data are a good fit with a hypothesized
distribution.
2. The 2 goodness of fit tests are always right tailed because the numerator in the
Chapter 11
Inferences on Two Samples
11.1 Inferences about Two Means: Dependent Samples
1. independent
2.
dependent
3. Since the researcher claims the mean of population 1, 1 , is less than the mean of
population 2, 2 , in matched pair data, the differenc
Chapter 10
Testing Claims Regarding a Parameter
10.1 The Language of Hypothesis Testing
1. A Type I error is the error of rejecting H 0 when in fact H 0 is true. A Type II error is the
error of not rejecting H 0 when in fact H1 is true.
2. The probability
Chapter 9
Estimating the Value of a Parameter Using Confidence Intervals
9.1 The Logic in Constructing Confidence Intervals about a Population
Mean Where the Population Standard Deviation is Known
1. The margin of error of a confidence interval of a param
Chapter 8
Sampling Distributions
8.1 Distribution of the Sample Mean
1. The sampling distribution of a statistic (such as the sample mean) is the probability
distribution for all possible values of the statistic computed from samples of fixed size, n.
2.
Chapter 7
The Normal Probability Distribution
7.1 Properties of the Normal Distribution
1. For the graph to be that of a probability density function,
(1) The area under the graph over all possible values of the random variable must equal 1;
(2) The graph
Chapter 5
Probability
5.1 Probability Rules
1. Empirical probability is based on the outcome of a probability experiment and is
approximately equal to the relative frequency of the event. Classical probability is based
on counting techniques and is equal
Chapter 4
Describing the Relation between Two Variables
4.1 Scatter Diagrams and Correlation
1. Univariate data measures the value of a single variable for each individual in the study.
Bivariate data measures values of two variables for each individual.
Chapter 3
Numerically Describing Data from One Variable
3.1 Measures of Central Tendency
1. A statistic is resistant if it is not sensitive to extreme data values. The median is resistant
because it is a positional measure of central tendency and increasi
Chapter 2
Organizing and Summarizing Data
2.1 Organizing Qualitative Data
1. Raw data are the data as originally collected, before they have been organized or coded.
2. number (or count); proportion (or percent)
3. It is a good idea to add up the frequenc