Hypotheses and Test Procedures
Some statements:
a. The mean strength of the pipe welds used in a nuclear power plant is
less than or equal to 100 lb/in2 .
a’. The mean strength of the pipe welds used in a nuclear power plant is
larger than 100 lb/in2 .
b. The proportion of certain power supply units need service during the
warranty period is at most 10%.
b’. The proportion of certain power supply units need service during the
warranty period is more than 10%.
c. The discharge-water temperature of the water used for cooling by a
power plant is normally distributed.
c’. The discharge-water temperature of the water used for cooling by a
power plant is exponentially distributed.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
1 / 10
Hypotheses and Test Procedures
Definition
The null hypothesis, denoted by H0 , is the claim that is initially assumed
to be true (the “prior belief” claim). The alternative hypothesis,
denoted by Ha , is the assertion that is contradictory to H0 .
The null hypothesis will be rejected in favor of the alternative hypothesis
only if sample evidence suggests that H0 is false. If the sample does not
strongly contradict H0 , we will continue to believe in the truth of the null
hypothesis.
Remark:
There are only two possible conclusions from a hypothesis testing: reject
H0 or fail to reject H0 ; or corresponding accept Ha or fail to
accept Ha .
We NEVER reject Ha .
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
2 / 10
Hypotheses and Test Procedures
Problem of interest: if the strength of the pipe welds used in a nuclear
power plant is assumed to be normally distributed , whether the mean
value is less than or equal to 100 lb/in2 .
H0 : µ ≤ 100 v.s. Ha : µ > 100.
After an experiment, we recorded 10 sample data:
101.9 100.4 101.2 100.9 101.7
with X = 101.10.
101.5 100.9 100.1 101.6 100.8
If we believe the sample mean should not exceed 100, then we should
reject H0 from this sample data.
However if we believe the sample mean could be a little bit larger than 100
but not exceed 102, then we should not reject H0 from this sample data.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
3 / 10
Hypotheses and Test Procedures
Problem of interest: whether the proportion of certain power supply units
need service during the warranty period is at most 10%.
H0 : p ≤ .1 v.s. Ha : p > .1.
After an experiment, we find that 3 units need service during the warranty
period among 20 samples. If the null hypothesis is true, then the average
number of units need service for this sample should not exceed
n · p = 20 · (.1) = 2.
If we believe the sample mean should not exceed 2, then we should reject
H0 from this sample data.
However if we believe the sample mean should not exceed 3, then we
should not reject H0 from this sample data.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
4 / 10
Hypotheses and Test Procedures
A test procedure is specified by the following:
1. A test statistic, a function of the sample data on which the decision
(reject H0 or do not rejece H0 ) is to be based;
2. A rejection region, the set of all test statistic values for which H0
will be rejected.
The null hypothesis will then be rejected if and only if the observed or
computed test statistic value falls in the rejection region.
Remark: The rejection region is more subjective.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
5 / 10
Hypotheses and Test Procedures
Definition
A type I error consists of rejecting the null hypothesis H0 when it is true,
traditionally denoted the probability of making type I error by α.
A type II error involves not rejecting H0 when it is false, traditionally
denoted the probability of making type I error by β.
Mathematically,
α = P(Type I Error) = P(reject H0 | H0 )
β = P(Type II Error) = P(fail to reject H0 | Ha )
Remark:
There are no error-free test procedures unless we have the information for
the whole population.
Therefore, we look for those test for which the probability of making either
type of error is small.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
6 / 10
Hypotheses and Test Procedures
Problem of interest: if the strength of the pipe welds used in a nuclear
power plant is assumed to be normally distributed , whether the mean
value is less than or equal to 100 lb/in2 .
H0 : µ = 100 v.s. Ha : µ > 100.
After an experiment, we recorded 10 sample data:
101.9 100.4 101.2 100.9 101.7
with X = 101.10.
101.5 100.9 100.1 101.6 100.8
If we choose the cut-off value to be 100, i.e., to reject H0 when X > 100,
what is the probability of making type I error and type II error?
However if we choose the cut-off value to be 102, i.e., to reject H0 when
X > 102, what is the probability of making type I error and type II error?
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
7 / 10
Hypotheses and Test Procedures
Problem of interest: whether the proportion of certain power supply units
need service during the warranty period is at most 10%.
H0 : p = .1 v.s. Ha : p > .1.
After an experiment, we find that 3 units need service during the warranty
period among 20 samples. If the null hypothesis is true, then the average
number of units need service for this sample should not exceed
n · p = 20 · (.1) = 2.
If we choose the cut-off value to be 2, i.e., to reject H0 when X > 2, what
is the probability of making type I error and type II error?
However if we choose the cut-off value to be 3, i.e., to reject H0 when
X > 3, what is the probability of making type I error and type II error?
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
8 / 10
Hypotheses and Test Procedures
Proposition
Suppose an experiment and a sample size are fixed and a test statistic is
chosen. Then decreasing the size of the rejection region to obtain a
smaller value of α results in a larger value of β for any particular
parameter value consistent with Ha .
Remark:
1. In other words, there is no rejection region that will simultaneously
make both α and all βs small.
2. First specify the largest value of α that can be tolerated and then find
a rejection region having that value of α rather than than anything
smaller. The resulting value of α is often referred to as the
significance level of the test.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
9 / 10
Hypotheses and Test Procedures
Problem of interest: if the strength of the pipe welds used in a nuclear
power plant is assumed to be normally distributed , whether the mean
value is less than or equal to 100 lb/in2 .
H0 : µ = 100 v.s. Ha : µ > 100.
After an experiment, we recorded 10 sample data:
101.9 100.4 101.2 100.9 101.7
with X = 101.10.
101.5 100.9 100.1 101.6 100.8
What is a level 5% test?
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008
10 / 10