Download Week 1: Descriptive Statistics

Topic 6: Introduction to
Hypothesis Testing
CEE 11 Spring 2002
Dr. Amelia Regan
These notes draw liberally from the class text, Probability and Statistics for
Engineering and the Sciences by Jay L. Devore, Duxbury 1995 (4th edition)
Additional material is taken from http://davidmlane.com/hyperstat/
Hypothesis testing



A statistical hypothesis is a claim either about the
value of a single population statistic or about the
values of several population statistics
A population statistic might be the mean diameter,
weight, time to failure, proportion of defective units,
etc.
One example of a hypothesis is that m = c, where c is
some constant. We might also hypothesize that
m1 > m2 or that p, a population proportion = c
Hypothesis testing
 In any hypothesis testing problem there are two
contradictory hypotheses under consideration these
are known as the Null and Alternative Hypotheses
 A test procedure is specified by the following:
A test statistic, a function of the sample data on which
the decision reject H0 or do not reject H0 is based.
A rejection region -- the set of all test statistic values for
this H0 will be rejected.
Hypothesis testing
 The null hypothesis will be rejected if and only if the observed or
computed test statistic value falls into the rejection region.
 A type I error consists of rejecting the null hypothesis when it is
true.
 A type II error consists of not rejecting the null hypothesis when it
is false.
 a = the probability of a type I error -- P(H0 is rejected when it is
in fact true)
 b = the probability of a type II error -- P(H0 is not rejected when it
is in fact false)
 a is often called the significance level of a test
Hypothesis testing
TRUTH (unknown)
Ho True
Ho False
Decision Reject
Type I Error
Correct Decision
Do not Reject Correct Decision Type II Error
 For a given rejection region – decreasing the probability of a
Type I error increases the probability of a Type II error. Similarly,
decreasing the probability of a Type I error increases the
probability of a Type II error.
Hypothesis testing
When an independent variable appears to have an
effect, it is very important to be able to state with
confidence that the effect was really due to the
variable and not just due to chance.
For instance, consider a hypothetical experiment on a
new antidepressant drug. Ten people suffering from
depression were sampled and treated with the new
drug (the experimental group); an additional 10
people were sampled from the same population and
were treated only with a placebo (the control group).
Hypothesis testing
After 12 weeks, the level of depression in all subjects
was measured and it was found that the mean level of
depression (on a 10-point scale with higher numbers
indicating more depression) was 4 for the
experimental group and 6 for the control group.
The most basic question that can be asked here is:
"How can one be sure that the drug treatment rather
than chance occurrences were responsible for the
difference between the groups?" It could be, that by
chance, the people who were randomly assigned to
the treatment group were initially somewhat less
depressed than those randomly assigned to the
control group.
Hypothesis testing
The null hypotheses in this case would be
H0: m1 = m2,meaning that the mean level of depression for
the two groups is not different in a statistically
significant way.
The alternative hypothesis would be
Ha: m1 < m2
Hypothesis testing -example
Problem 12, Page 321
A new design on the braking system on a certain type of
new car has been proposed. For the current system,
the true average braking distance at 40 mph (under
specified conditions) is known to be 120 feet.
It is proposed that the new design be implemented only
if sample data strongly indicates that the new design
leads to a reduction in the braking distance.
a) Define the parameter of interest
Mean braking distance under the new system
Hypothesis testing -example
b) Suppose that braking distance for the new system
is normally distributed with s = 10.
Let X represent the mean braking distance for a
sample of 36 observations. Which of the following
rejection regions is appropriate?
R1   X : X  124.8
R2   X : X  115.2
R2   X : X  125.13 or X  114.87
To answer this, we need to identify the null
and alternative hypothesis.
H o : m  120, H a : m  120
Hypothesis testing -example
b) The test statistic is normally distributed
N(120,1.667) therefore, the tests below correspond
to a values of
R1   X : X  124.8
R2   X : X  115.2
a =0.002
R2   X : X  125.13 or X  114.87
Hypothesis testing -example
c) The significance level =
115.2  120
a  P( z 
)  0.002
1.667
To obtain a = 0.001 we need to change the rejection
region so that it begins with z = -3.08
z
xm
s
 x  zs  m 
x  3.08(1.667)  120  114.87
Hypothesis testing -example
d) The probability that the new design is not
implemented when its true average braking
distance is actually 115 and the appropriate region
from b) is used is
b  P( z 
xm
s
 115.20)
Draw a picture!
115.20  115
b  P( z 
)  P( z  0.12)  0.4522
1.667
Sampling Distributions
When s for a population is known, then the mean of a sample
from the population is normally distributed
X
s
X m
N (m ,
)Z 
 s 
n


 n
However, we rarely know s. Instead, we must estimate the
standard deviation from the sample data. In that case, the
mean follows a similar but different distribution (known as
the student’s t distribution or the t distribution)
X m
t
 s 


 n
Tests Statistics and
Rejection Regions
CASE I: A Normal Population with known standard deviation
Null hypothesis
H o : m =mo
Test statistic value
z
x  mo
s
n
Alternative hypothesis
H a : m  mo
H a : m  mo
H a : m  mo
Rejection region for level a test
z  za
z  za
z  za
2
or z  za
2
Tests Statistics and
Rejection Regions
CASE II: Large Sample Test (n >30 – the average is
approximately normal and s is a good approximation for s)
Null hypothesis
H o : m =mo
Test statistic value
x  mo
z
s
n
Alternative hypothesis
H a : m  mo
H a : m  mo
H a : m  mo
Rejection region for level a test
z  za
z  za
z  za
2
or z  za
2
Tests Statistics and
Rejection Regions
CASE III: A Normal Population with unknown standard
deviation
Null hypothesis
H o : m =mo
Test statistic value
Alternative hypothesis
x  mo
t
s
n
Rejection region for level a test
H a : m  mo
t  tn 1,a
H a : m  mo
t  tn 1,a
H a : m  mo
t  tn 1,a
or t  tn 1,a
2
2