Hypothesis (Goodness of Fit) Test
for Proportions of a Multinomial Population
Tests of Goodness of Fit
Goodness of Fit Test: A Multinomial Population
n Goodness of Fit Test: Poisson
and Normal Distributions
1. Set up the null and alternative hypotheses.
F distribution
f-test
Tim Low
February 18, 2015
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F distribution
A very important set of hypothesis tests in statistics is the two
sample t-test but this test cannot be conducted until we know
whether the population variances are equal - hence we do
February 25, 2015
Checking the Required Conditions
Violation of the Required Conditions
The F-test of the analysis of variance requires that the
random variable be normally distributed with equal
variances. The normality requirement is easily checked
grap
March 4, 2015
Two-Factor Analysis of Variance
Two-Factor Analysis of Variance
At the start of this course we addressed problems where the
data were generated from single-factor experiments.
In factorial experiments, we can examine the effect on the
respon
February 19, 2015
Example 1
IDENTIFY
The null hypothesis in this case is:
H0: 1 = 2 = 3
Test Statistic
Since 1 = 2 = 3 is of interest to us, a statistic that
measures the proximity of the sample means to each other
would also be of interest.
i.e. there ar
March 4, 2015
Example 17
Example 17
A two factor analysis of variance experiment was performed
with a = 3, b = 4 and r = 20. The following sums of squares
were computed:
SS(Total) = 42450
SS(A) = 1560
SS(B) = 2880
SS(AB) = 7605
a) Determine the ANOVA tabl
February 26, 2015
Example 10
Example 10
A pharmaceutical company has recently developed four new
drugs to lower cholesterol levels.
To determine whether any differences exist in their benets,
an experiment was organized. The company selected 25
groups of
February 25, 2015
Analysis of Variance Experimental Designs
Independent Samples and Blocks
Experimental design determines which analysis of variance
technique we use.
Similar to the matched pairs experiment, a randomized
block design experiment reduces th
March 4, 2015
Example 12
Example 13
The accompanying table presents data on yields relating to resistance to stain for
three materials (M1, M2 and M3) treated with four chemicals in a randomized block
design. (A low value indicates good strain resistance)
February 18, 2015
Analysis of Variance
Analysis of variance is a technique that allows us to
compare two or more populations of interval data.
Analysis of Variance
Analysis of variance is:
an extremely powerful and widely used procedure.
a procedure whi
t-test
Assumptions
Tim Low
February 16, 2015
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The two-sample t-test
In real problems it is virtually always the case that the values of
the population variances are unknown. If the sample sizes are
large, the central limit theorem allows us to still
Time series
Forecasting
Tim Low
April 20, 2015
1/1
Forecasting
In forecasting we use data from the past in predicting the future
value of the variable of interest, such as appliance sales, natural
gas consumption, or the thickness of the ozone layer.
2/1
February 23, 2015
ANOVA and t-tests of 2 means
Completely Randomised Design
Why do we need the analysis of variance? Why not test every pair of
means? For example say k = 6. There are C26 = 6(5)/2= 14 different
pairs of means.
1&2 1&3 1&4 1&5 1&6
2&3 2&4
March 11, 2015
Coefficient of Determination
Coefficient of Determination
Tests thus far have shown if a linear relationship exists; it is
also useful to measure the strength of the relationship. This
is done by calculating the coefcient of determination R
March 9, 2015
Regression Analysis
Our problem objective is to analyze the relationship
between interval variables; regression analysis is the rst
tool we will study.
Simple Linear Regression
and Correlation
Regression analysis is used to predict the value
Determining When to Add or Delete Variables
Regression Analysis: Model Building
Variable Selection Procedures
Multiple Regression Approach to Experimental Design
The F Test is based on a determination of the amount of
reduction in the error sum of squares
Time series
Seasonal Indexes
Tim Low
April 16, 2015
1/1
Seasonal Indexes
Depending on the nature of the time series, strong seasonal
patterns may be present for certain months or quarters.
Depending on the nature of the time series, strong seasonal
patter
Time series
Smoothing techniques
Tim Low
April 14, 2015
1/1
Components of a Time Series
2/1
Moving averages
The moving average replaces the original time series with
another series, each point of which is the center of and the
average of N points from the
April 16, 2015
Regression Analysis
Model Building
Regression analysis is one of the most powerful and
commonly used techniques in statistics; it allows us to create
mathematical models that realistically describe relationships
between the dependent variab
Variable Selection: Best-Subsets Regression
Variable Selection: Best-Subsets Regression
The three preceding procedures are one-variable-at-atime methods offering no guarantee that the best
model for a given number of variables will be found.
Example: PGA
April 16, 2015
Time Series Models
Time Series Models
There are many models that are available for forecasting a time
series.
Alternatively, we assume that the relationship is multiplicative
In order to forecast, we assume that time series are made up of
April 16, 2015
Time Series Analysis
A variable measured over time (in sequential order) is called
a time series. From this data, we analyze it to detect patterns
that will enable us to forecast future values of the variable.
Time Series Analysis and
Forec
March 18, 2015
Qualitative Independent variables
Example 3
In many situations we must work with qualitative independent
variables such as gender (male, female), method of payment (cash,
cheque, credit card), etc.
A software rm are
interested in determinin
Example 2
Example 2, cont.
An efficiency expert has studied 12 employees
who perform similar assembly tasks, recording
productivity (units per hour), number of years of
experience, and which one of three popular
assembly methods the individual has chosen
March 11, 2015
Regression Diagnostics
Residual Analysis
There are three conditions that are required in order to
perform a regression analysis. These are:
The error variable must be normally distributed,
The error variable must have a constant variance,
March 11, 2015
Multiple Regression
The simple linear regression model was used to analyze how
one interval variable (the dependent variable y) is related to
one other interval variable (the independent variable x).
Multiple Regression
Multiple regression
STA2020F 2009
Class Test 1
Date: 27th March 2009
Total marks: 70
Time: 1 hour 30 minutes
Instructions: Answer all the questions in your blue answer book. Your
answer should be T or F for each question in section A and a written
answer for section B. There
UNIVERSITY OF CAPE TOWN
DEPARTMENT OF STATISTICAL SCIENCES
STA2020F
TEST 2
Hannah Gerber
Time: 1 hour 30 minutes
Internal examiner(s):
19 April 2010
Total marks: 50
Date:
Total number of questions: 3
Total number of pages: 17
Instructions: Answer all ques