Confidence Intervals for the Variance of a Normal
Population
Confidence Intervals for the Variance of a Normal
Population
Example (a variant of Problem 62, Ch5)
The total time for manufacturing a certain component is known to
have a normal distribution. However, the mean µ and variance σ 2
for the normal distribution are unknown. After an experiment in
which we manufactured 10 components, we recorded the sample
time which is given as follows:
1
2
3
4
5
time 63.8 60.5 65.3 65.7 61.9
with
6
7
8
9
10
time 68.2 68.1 64.8 65.8 65.4
X = 64.95, s = 2.42
Confidence Intervals for the Variance of a Normal
Population
Example (a variant of Problem 62, Ch5)
The total time for manufacturing a certain component is known to
have a normal distribution. However, the mean µ and variance σ 2
for the normal distribution are unknown. After an experiment in
which we manufactured 10 components, we recorded the sample
time which is given as follows:
1
2
3
4
5
time 63.8 60.5 65.3 65.7 61.9
with
6
7
8
9
10
time 68.2 68.1 64.8 65.8 65.4
X = 64.95, s = 2.42
What is a 95% confidence for the population variance σ 2 ?
Confidence Intervals for the Variance of a Normal
Population
Confidence Intervals for the Variance of a Normal
Population
Theorem
Let X1 , X2 , . . . , Xn be a random sample from a distribution with
mean µ and variance σ 2 . Then the random variable
P
(Xi − X )2
(n − 1)S 2
=
σ2
σ2
has s chi-squared (χ2 ) probability distribution with n − 1 degrees
of freedom (df).
Confidence Intervals for the Variance of a Normal
Population
Confidence Intervals for the Variance of a Normal
Population
Confidence Intervals for the Variance of a Normal
Population
Confidence Intervals for the Variance of a Normal
Population
Notation
Let χ2α,ν , called a chi-squared critical value, denote the number
on the measurement axis such that α of the area under the
chi-squared curve with ν df lies to the right of χ2α,ν .
Confidence Intervals for the Variance of a Normal
Population
Notation
Let χ2α,ν , called a chi-squared critical value, denote the number
on the measurement axis such that α of the area under the
chi-squared curve with ν df lies to the right of χ2α,ν .
Confidence Intervals for the Variance of a Normal
Population
Confidence Intervals for the Variance of a Normal
Population
Proposition
A 100(1 − α)% confidence interval for the variance σ 2 of a
normal population has lower limit
(n − 1)s 2 /χ2α ,n−1
2
and upper limit
(n − 1)s 2 /χ21− α ,n−1
2
A confidence interval for σ has lower and upper limits that are
the square roots of the corresponding limits in the interval for σ 2 .
Hypotheses and Test Procedures
Hypotheses and Test Procedures
Some statements:
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 .
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 .
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%.
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%.
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.
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.
Hypotheses and Test Procedures
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.
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 .
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 .
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.
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 believe the sample mean should not exceed 100, then we
should reject H0 from this sample data.
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.
Hypotheses and Test Procedures
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%.
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.
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.
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.
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.
Hypotheses and Test Procedures
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.
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.
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?
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?
Hypotheses and Test Procedures
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.
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?
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?
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?