Hypotheses and Test Procedures
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 )
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.
Hypotheses and Test Procedures
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
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 (assume the true population mean is µ = 101)?
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 (assume the true population mean is µ = 101)?
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 (assume the true population mean is µ = 101)?
Hypotheses and Test Procedures
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 .
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.
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.
Hypotheses and Test Procedures
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
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?