Hypothesis Testing
A statistical Test is defined by
1. Choosing a statistic (called the test
statistic)
2. Dividing the range of possible values for
the test statistic into two parts
The Acceptance Region
The Critical Region
To perform a statistical Tes
Summary of Tests
Confidence Limits
Proportions and means
One sample Tests
Z test for a proportion
Test statistic
z
p p0
p0 1 p0
n
Null Hypothesis
Alt. Hypothesis
H0: p p0
Critical Region
z < -z or z > z
H0: p = p0
H0: p > p0
z > z
H0: p < p0
z < -z
Confi
The p-value approach to
Hypothesis Testing
In hypothesis testing we need
1. A test statistic
2. A Critical and Acceptance region
for the test statistic
The Critical Region is set up under the sampling
distribution of the test statistic.
Area = (0.05 or 0.
Comparing k Populations
Means One way Analysis of
Variance (ANOVA)
The F test for comparing k means
Situation
We have k normal populations
Let i and denote the mean and standard
deviation of population i.
i = 1, 2, 3, k.
Note: we assume that the stand
Comparing k Populations
Means One way Analysis of
Variance (ANOVA)
Example
Inthisexamplewearelookingattheweightgains(grams)
forratsundersixdietsdifferinginlevelofprotein(High
orLow) andsourceofprotein(Beef,Cereal,orPork).
Ten test animals for each diet
D
Comparing k Populations
Means One way Analysis of
Variance (ANOVA)
Example
Inthisexamplewearelookingattheweight
gains(grams)forratsundersixdietsdiffering
inlevelofprotein(HighorLow) andsource
ofprotein(Beef,Cereal,orPork).
Ten test animals for each diet
Comparing Populations
Proportions and means
The sampling distribution of differences of Normal
Random Variables
If X and Y denote two independent normal random
variables, then :
D = X Y is normal with
mean D X Y
2
2
standard deviation D X Y
Comparing prop
Statistical Inference
Making decisions regarding the
population base on a sample
Decision Types
Estimation
Deciding on the value of an unknown parameter
Hypothesis Testing
Deciding a statement regarding an unknown parameter
is true of false
All decis
Hypothesis Testing
To define a statistical Test we
1. Choose a statistic (called the test statistic)
2. Divide the range of possible values for the
test statistic into two parts
The Acceptance Region
The Critical Region
To perform a statistical Test we
STAT 244.3
Some Sample Questions for Term Test #1 Winter 2012
PLEASE READ IMPORTANT TEST INFORMATION
(i) You may use one two-sided 8.5x11 sheet of formulas, etc. Textbook is NOT allowed.
(ii) Non-programmable, non-graphing calculators are allowed.
(iii) I
STAT 244.3 Written Assignment #1
Due: by 3:00 PM, Friday, January 20, 2012
Note: You may turn the assignment in earlier (e.g. in class or at next Thursdays lab)
or place it in the In box outside of my office (239 McLean Hall).
PROBLEM: The data given on t
Sampling Theory
Determining the distribution of Sample
statistics
Sampling Theory
sampling distributions
Note:It is important to recognize the dissimilarity
(variability) we should expect to see in various
samples from the same population.
It is importan
Stats 244.3
Elementary Statistical Concepts
Instructor:
W.H.Laverty
Office:
235 McLean Hall
Phone:
966-6096
Lectures:
Evaluation:
M Tu W Th F 10:30am -11:50am
Arts 200
Lab: Tu W Th 12:00 - 12:50 Arts 200
Assignments, Labs, Term tests - 40%
Each Friday Ter
Logistic regression
Recall the simple linear regression model:
y = 0 + 1x +
where we are trying to predict a continuous
dependent variable y from a continuous
independent variable x.
This model can be extended to Multiple linear
regression model:
y = 0 +
Linear Regression
Hypothesis testing and Estimation
The equation for the least squares line
n
Let
2
S xx xi x
i
1
n
S yy yi y
2
i
1
n
S xy xi x yi y
i
1
Computing Formulae:
n
2
xi
n
n
2
i 1
2
S xx xi x xi
n
i
1
i
1
2
n
yi
n
n
2
S yy yi y yi2 i 1
Linear Regression
Hypothesis testing and Estimation
Assume that we have collected data on two
variables X and Y. Let
(x1, y1) (x2, y2) (x3, y3) (xn, yn)
denote the pairs of measurements on the on
two variables X and Y for n cases in a sample
(or populatio
STAT 244.3
Solution to Sample Questions related to Term Test #3 Material A
PROBLEM SET A: Seven randomly selected women weighing between 60.0 and 69.9
kilograms were placed on a cabbage soup diet for a period of four weeks. Their weight losses
in kilogram
STAT 244.3 Written Assignment #3
Due: Friday, February 3 by 2:30 PM
PROBLEM A: How physically active are students? A random sample of students was obtained
by sampling student numbers from a list of such numbers for current students. Each student
indicate
STATISTICS 244.3
WRITTEN ASSIGNMENT #6
Due: Problems C and D on March 21 by 2:30 PM
Problems A and B are for practice. Solutions will be posted.
Note: Term Test #3 is on Thursday, March 22 at 1:00 PM
PROBLEM A: A simple random sample of 75 students who li
[STAT 244.3 Written Assignment #2]
Name
Student Number
STAT 244.3
Written Assignment #2
Due Date: by 3:30 PM, Tuesday, January 24, 2012
PLEASE NOTE: Provide answers rounded correctly to two places of decimal
(where 2 or more places of decimal are determin
Sampling Theory
Determining the distribution of Sample
statistics
Sampling Theory
sampling distributions
Note:It is important to recognize the dissimilarity
(variability) we should expect to see in various
samples from the same population.
It is importan
Hypothesis Testing
A statistical Test is defined by
1. Choosing a statistic (called the test
statistic)
2. Dividing the range of possible values for
the test statistic into two parts
The Acceptance Region
The Critical Region
To perform a statistical Tes
The Normal Probability Distribution
Points of
Inflection
3 2
2
3
Main characteristics of the Normal Distribution
Bell Shaped, symmetric
Points of inflection on the bell shaped curve are at
and + That is one standard deviation
from the mean
Area
Stats 244 - Asst 3 Solutions
a) Construct the stem and leaf display for Math proficiency for students who had
completed grade 13.
5
3567
6
0119
7
0123344445567
8
05
9
0
b) Construct the stem and leaf display for Math proficiency for students who had
compl
Stats 244 Assignment 2 - Solutions
For each of the three drugs A, B and C
a) Compute the mean and the standard deviation.
Drug A
n
x=
x
i =1
i
n
=
209.2
= 10.46 , s =
20
n
xi
n
2
xi i =1
2
2431.72 (209.2 )
2
n
i =1
n 1
=
20 = 3.58
19
Drug B
n
x=
x
i
Stats 244 Assignment 1
(Not to be handed in)
I.
In the following study, 20 manuscripts of an unknown author were found. Some of the evidence proved
that these unknown manuscripts were either written by author A or author B. To decide between these two
ind
o q _Sroct5 9% pd? -
Stats 244.3 Test 6
Monday, April 7
Open Book (Select answers cl
i. The pre-test score (x) and
ii. The nal mark (y) on the calculus course.
3 S3. 335. c
Choices for 1. to 3.
a) 3236.4 b) 3256.6 c) 3306.3 (1) 3332.4 e) 3354.6
1) 3438.
SCHNNED
Stats 244. 3 Test 6
Thursday . March 3 l
Open Book (Select answers closest to the correct answers)
I. In the following study n = IO students, enrolled In a 151 year calculus course, were
given pre-test in high school mathematics prior to taking th
Assignment 3 Stats 244.3(02)
Table I.1 gives scores on a university math proficiency test for n = 24 Ontario
students who had completed grade 13. Table I.2 gives scores on the same university math
proficiency test for m = 30 Ontario students who were requ