Applied Statistics I
Liang Zhang
Department of Mathematics, University of Utah
July 21, 2008
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 1 / 22

Hypotheses and Test Procedures
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 2 / 22
Some statements: a. The mean strength of the pipe welds used in a nuclear power plant is less than or equal to 100 lb/in
2
.
a’. The mean strength of the pipe welds used in a nuclear power plant is larger than 100 lb/in
2
.
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
Some statements:
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 2 / 22 a. The mean strength of the pipe welds used in a nuclear power plant is less than or equal to 100 lb/in
2
.
a’. The mean strength of the pipe welds used in a nuclear power plant is larger than 100 lb/in
2
.
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
Some statements: a. The mean strength of the pipe welds used in a nuclear power plant is less than or equal to 100 lb/in
2
.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 2 / 22 a’. The mean strength of the pipe welds used in a nuclear power plant is larger than 100 lb/in
2
.
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
Some statements: a. The mean strength of the pipe welds used in a nuclear power plant is less than or equal to 100 lb/in
2
.
a’. The mean strength of the pipe welds used in a nuclear power plant is larger than 100 lb/in
2
.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 2 / 22 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
Some statements: a. The mean strength of the pipe welds used in a nuclear power plant is less than or equal to 100 lb/in
2
.
a’. The mean strength of the pipe welds used in a nuclear power plant is larger than 100 lb/in
2
.
b. The proportion of certain power supply units need service during the warranty period is at most 10%.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 2 / 22 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
Some statements: a. The mean strength of the pipe welds used in a nuclear power plant is less than or equal to 100 lb/in
2
.
a’. The mean strength of the pipe welds used in a nuclear power plant is larger than 100 lb/in
2
.
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%.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 2 / 22 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
Some statements: a. The mean strength of the pipe welds used in a nuclear power plant is less than or equal to 100 lb/in
2
.
a’. The mean strength of the pipe welds used in a nuclear power plant is larger than 100 lb/in
2
.
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.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 2 / 22 c’. The discharge-water temperature of the water used for cooling by a power plant is exponentially 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/in
2
.
a’. The mean strength of the pipe welds used in a nuclear power plant is larger than 100 lb/in
2
.
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 2 / 22

Hypotheses and Test Procedures
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 3 / 22
Definition
The null hypothesis , denoted by H
0
, is the claim that is initially assumed to be true (the “prior belief” claim). The alternative hypothesis , denoted by H a
, is the assertion that is contradictory to H
0
.
The null hypothesis will be rejected in favor of the alternative hypothesis only if sample evidence suggests that H
0 is false. If the sample does not strongly contradict H
0
, we will continue to believe in the truth of the null hypothesis.
Remark:
There are only two possible conclusions from a hypothesis testing: reject
H
0 or fail to reject H
0
; or corresponding accept H a or fail to accept H a
.
We NEVER reject H a
.

Hypotheses and Test Procedures
Definition
The null hypothesis , denoted by H
0
, is the claim that is initially assumed to be true (the “prior belief” claim). The alternative hypothesis , denoted by H a
, is the assertion that is contradictory to H
0
.
The null hypothesis will be rejected in favor of the alternative hypothesis only if sample evidence suggests that H
0 is false. If the sample does not strongly contradict H
0
, we will continue to believe in the truth of the null hypothesis.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 3 / 22
Remark:
There are only two possible conclusions from a hypothesis testing: reject
H
0 or fail to reject H
0
; or corresponding accept H a or fail to accept H a
.
We NEVER reject H a
.

Hypotheses and Test Procedures
Definition
The null hypothesis , denoted by H
0
, is the claim that is initially assumed to be true (the “prior belief” claim). The alternative hypothesis , denoted by H a
, is the assertion that is contradictory to H
0
.
The null hypothesis will be rejected in favor of the alternative hypothesis only if sample evidence suggests that H
0 is false. If the sample does not strongly contradict H
0
, we will continue to believe in the truth of the null hypothesis.
Remark:
There are only two possible conclusions from a hypothesis testing: reject
H
0 or fail to reject H
0
; or corresponding accept H a or fail to accept H a
.
We NEVER reject H a
.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 3 / 22

Hypotheses and Test Procedures
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 4 / 22
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/in
2
.
H
0
: µ ≤ 100 v.s.
H a
: 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 H
0 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 H
0 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/in
2
.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 4 / 22
H
0
: µ ≤ 100 v.s.
H a
: 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 H
0 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 H
0 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/in
2
.
H
0
: µ ≤ 100 v.s.
H a
: µ > 100.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 4 / 22
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 H
0 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 H
0 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/in
2
.
H
0
: µ ≤ 100 v.s.
H a
: 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
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 4 / 22
If we believe the sample mean should not exceed 100, then we should reject H
0 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 H
0 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/in
2
.
H
0
: µ ≤ 100 v.s.
H a
: 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 H
0 from this sample data.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 4 / 22
However if we believe the sample mean could be a little bit larger than 100 but not exceed 102, then we should not reject H
0 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/in
2
.
H
0
: µ ≤ 100 v.s.
H a
: 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 H
0 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 H
0 from this sample data.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 4 / 22

Hypotheses and Test Procedures
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 5 / 22
Problem of interest: whether the proportion of certain power supply units need service during the warranty period is at most 10%.
H
0
: p ≤ .
1 v.s.
H a
: 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
H
0 from this sample data.
However if we believe the sample mean should not exceed 3, then we should not reject H
0 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%.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 5 / 22
H
0
: p ≤ .
1 v.s.
H a
: 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
H
0 from this sample data.
However if we believe the sample mean should not exceed 3, then we should not reject H
0 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%.
H
0
: p ≤ .
1 v.s.
H a
: p > .
1.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 5 / 22
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
H
0 from this sample data.
However if we believe the sample mean should not exceed 3, then we should not reject H
0 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%.
H
0
: p ≤ .
1 v.s.
H a
: 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.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 5 / 22
If we believe the sample mean should not exceed 2, then we should reject
H
0 from this sample data.
However if we believe the sample mean should not exceed 3, then we should not reject H
0 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%.
H
0
: p ≤ .
1 v.s.
H a
: 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
H
0 from this sample data.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 5 / 22
However if we believe the sample mean should not exceed 3, then we should not reject H
0 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%.
H
0
: p ≤ .
1 v.s.
H a
: 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
H
0 from this sample data.
However if we believe the sample mean should not exceed 3, then we should not reject H
0 from this sample data.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 5 / 22

Hypotheses and Test Procedures
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 6 / 22
A test procedure is specified by the following:
1. A test statistic , a function of the sample data on which the decision
(reject H
0 or do not rejece H
0
) is to be based;
2. A rejection region , the set of all test statistic values for which H
0 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
A test procedure is specified by the following:
1. A test statistic , a function of the sample data on which the decision
(reject H
0 or do not rejece H
0
) is to be based;
2. A rejection region , the set of all test statistic values for which H
0 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.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 6 / 22
Remark: The rejection region is more subjective.

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 H
0 or do not rejece H
0
) is to be based;
2. A rejection region , the set of all test statistic values for which H
0 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 6 / 22

Hypotheses and Test Procedures
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 7 / 22
Definition
A type I error consists of rejecting the null hypothesis H
0 when it is true, traditionally denoted the probability of making type I error by α .
A type II error involves not rejecting H
0 when it is false, traditionally denoted the probability of making type I error by β .
Mathematically,
α = P (Type I Error) = P (reject H
0
| H
0
)
β = P (Type II Error) = P (fail to reject H
0
| H a
)
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
Definition
A type I error consists of rejecting the null hypothesis H
0 when it is true, traditionally denoted the probability of making type I error by α .
A type II error involves not rejecting H
0 when it is false, traditionally denoted the probability of making type I error by β .
Mathematically,
α = P (Type I Error) = P (reject H
0
| H
0
)
β = P (Type II Error) = P (fail to reject H
0
| H a
)
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 7 / 22
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
Definition
A type I error consists of rejecting the null hypothesis H
0 when it is true, traditionally denoted the probability of making type I error by α .
A type II error involves not rejecting H
0 when it is false, traditionally denoted the probability of making type I error by β .
Mathematically,
α = P (Type I Error) = P (reject H
0
| H
0
)
β = P (Type II Error) = P (fail to reject H
0
| H a
)
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 7 / 22

Hypotheses and Test Procedures
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 8 / 22
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/in 2 .
H
0
: µ = 100 v.s.
H a
: 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 H
0 when what is the probability of making type I error and type II error ?
X > 100,
However if we choose the cut-off value to be 102, i.e., to reject H
0 when
X > 102, 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/in 2 .
H
0
: µ = 100 v.s.
H a
: 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
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 8 / 22
If we choose the cut-off value to be 100, i.e., to reject H
0 when what is the probability of making type I error and type II error ?
X > 100,
However if we choose the cut-off value to be 102, i.e., to reject H
0 when
X > 102, 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/in 2 .
H
0
: µ = 100 v.s.
H a
: 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 H
0 when what is the probability of making type I error and type II error ?
X > 100,
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 8 / 22
However if we choose the cut-off value to be 102, i.e., to reject H
0 when
X > 102, 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/in 2 .
H
0
: µ = 100 v.s.
H a
: 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 H
0 when what is the probability of making type I error and type II error ?
X > 100,
However if we choose the cut-off value to be 102, i.e., to reject H
0 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 8 / 22

Hypotheses and Test Procedures
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 9 / 22
Problem of interest: whether the proportion of certain power supply units need service during the warranty period is at most 10%.
H
0
: p = .
1 v.s.
H a
: 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 H
0 when is the probability of making type I error and type II error ?
X > 2, what
However if we choose the cut-off value to be 3, i.e., to reject H
0 when
X > 3, 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%.
H
0
: p = .
1 v.s.
H a
: 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.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 9 / 22
If we choose the cut-off value to be 2, i.e., to reject H
0 when is the probability of making type I error and type II error ?
X > 2, what
However if we choose the cut-off value to be 3, i.e., to reject H
0 when
X > 3, 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%.
H
0
: p = .
1 v.s.
H a
: 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 H
0 when is the probability of making type I error and type II error ?
X > 2, what
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 9 / 22
However if we choose the cut-off value to be 3, i.e., to reject H
0 when
X > 3, 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%.
H
0
: p = .
1 v.s.
H a
: 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 H
0 when is the probability of making type I error and type II error ?
X > 2, what
However if we choose the cut-off value to be 3, i.e., to reject H
0 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 9 / 22

Hypotheses and Test Procedures
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 10 / 22
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 H a
.
Remark:
1. In other words, there is no rejection region that will simultaneously make both α and all β s 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
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 H a
.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 10 / 22
Remark:
1. In other words, there is no rejection region that will simultaneously make both α and all β s 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
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 H a
.
Remark:
1. In other words, there is no rejection region that will simultaneously make both α and all β s small.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 10 / 22
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
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 H a
.
Remark:
1. In other words, there is no rejection region that will simultaneously make both α and all β s 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 10 / 22

Hypotheses and Test Procedures
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 11 / 22
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/in
2
.
H
0
: µ = 100 v.s.
H a
: 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?

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/in
2
.
H
0
: µ = 100 v.s.
H a
: 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
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 11 / 22
What is a level 5% test?

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/in
2
.
H
0
: µ = 100 v.s.
H a
: 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 11 / 22

Test about a Population Mean
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 12 / 22
Example:
To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of 10 welds is selected, and tests are conducted on each weld in the sample. The sample data is recorded as follows
101.9
100.4
101.2
100.9
101.7
101.5
100.9
100.1
101.6
100.8
with X = 101 .
10 .
It is known that the weld strength is normally distributed with mean µ and standard deviation σ = 2 . If the specifications state that the mean strength should be equal to 100 lb/in
2
, shall we accept that the pipe welds meet the specifications with significance level .05?

Test about a Population Mean
Example:
To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of 10 welds is selected, and tests are conducted on each weld in the sample. The sample data is recorded as follows
101.9
100.4
101.2
100.9
101.7
101.5
100.9
100.1
101.6
100.8
with X = 101 .
10 .
It is known that the weld strength is normally distributed with mean µ and standard deviation σ = 2 . If the specifications state that the mean strength should be equal to 100 lb/in
2
, shall we accept that the pipe welds meet the specifications with significance level .05?
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 12 / 22

Test about a Population Mean
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 13 / 22
1. Parameter of interest: µ = population average strength.
2. Null hypothesis: H
0
: µ = µ
0
= 100.
3. Alternative hypothesis: H a
: µ = 100.
4. Test statistic value: z =
¯ − µ
σ/
√ n
0
=
¯ − 100
2 /
√ n
5. Rejection region: z ≥ z
.
025 or z ≤ −
6. Substituting n = 10 and ¯ = 101 .
10, z
.
025
, where z
.
025
= 1 .
96.
z =
101 .
10 − 100
2 / 10
= 1 .
74
7. Since − 1 .
96 < 1 .
74 < 1 .
96, i.e., the value of the test statistic does not fall in the rejection region ( −∞ , 1 .
96) ∪ (1 .
96 , ∞ ), we can not reject H
0 at significance level .05.

Test about a Population Mean
1. Parameter of interest: µ = population average strength.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 13 / 22
2. Null hypothesis: H
0
: µ = µ
0
= 100.
3. Alternative hypothesis: H a
: µ = 100.
4. Test statistic value: z =
¯ − µ
σ/
√ n
0
=
¯ − 100
2 /
√ n
5. Rejection region: z ≥ z
.
025 or z ≤ −
6. Substituting n = 10 and ¯ = 101 .
10, z
.
025
, where z
.
025
= 1 .
96.
z =
101 .
10 − 100
2 / 10
= 1 .
74
7. Since − 1 .
96 < 1 .
74 < 1 .
96, i.e., the value of the test statistic does not fall in the rejection region ( −∞ , 1 .
96) ∪ (1 .
96 , ∞ ), we can not reject H
0 at significance level .05.

Test about a Population Mean
1. Parameter of interest:
2. Null hypothesis:
µ = population average strength.
H
0
: µ = µ
0
= 100.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 13 / 22
3. Alternative hypothesis:
4. Test statistic value:
H a
: µ = 100.
z =
¯ − µ
σ/
√ n
0
=
¯ − 100
2 /
√ n
5. Rejection region: z ≥ z
.
025 or z ≤ −
6. Substituting n = 10 and ¯ = 101 .
10, z
.
025
, where z
.
025
= 1 .
96.
z =
101 .
10 − 100
2 / 10
= 1 .
74
7. Since − 1 .
96 < 1 .
74 < 1 .
96, i.e., the value of the test statistic does not fall in the rejection region ( −∞ , 1 .
96) ∪ (1 .
96 , ∞ ), we can not reject H
0 at significance level .05.

Test about a Population Mean
1. Parameter of interest: µ = population average strength.
2. Null hypothesis: H
0
: µ = µ
0
= 100.
3. Alternative hypothesis: H a
: µ = 100.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 13 / 22
4. Test statistic value: z =
¯ − µ
σ/
√ n
0
=
¯ − 100
2 /
√ n
5. Rejection region: z ≥ z
.
025 or z ≤ −
6. Substituting n = 10 and ¯ = 101 .
10, z
.
025
, where z
.
025
= 1 .
96.
z =
101 .
10 − 100
2 / 10
= 1 .
74
7. Since − 1 .
96 < 1 .
74 < 1 .
96, i.e., the value of the test statistic does not fall in the rejection region ( −∞ , 1 .
96) ∪ (1 .
96 , ∞ ), we can not reject H
0 at significance level .05.

Test about a Population Mean
1. Parameter of interest: µ = population average strength.
2. Null hypothesis: H
0
: µ = µ
0
= 100.
3. Alternative hypothesis: H a
: µ = 100.
4. Test statistic value: z =
¯ − µ
σ/
√ n
0
=
¯ − 100
2 /
√ n
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 13 / 22
5. Rejection region: z ≥ z
.
025 or z ≤ −
6. Substituting n = 10 and ¯ = 101 .
10, z
.
025
, where z
.
025
= 1 .
96.
z =
101 .
10 − 100
2 / 10
= 1 .
74
7. Since − 1 .
96 < 1 .
74 < 1 .
96, i.e., the value of the test statistic does not fall in the rejection region ( −∞ , 1 .
96) ∪ (1 .
96 , ∞ ), we can not reject H
0 at significance level .05.

Test about a Population Mean
1. Parameter of interest: µ = population average strength.
2. Null hypothesis: H
0
: µ = µ
0
= 100.
3. Alternative hypothesis: H a
: µ = 100.
4. Test statistic value: z =
¯ − µ
σ/
√ n
0
=
¯ − 100
2 /
√ n
5. Rejection region: z ≥ z
.
025 or z ≤ − z
.
025
, where z
.
025
= 1 .
96.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 13 / 22
6. Substituting n = 10 and ¯ = 101 .
10, z =
101 .
10 − 100
2 / 10
= 1 .
74
7. Since − 1 .
96 < 1 .
74 < 1 .
96, i.e., the value of the test statistic does not fall in the rejection region ( −∞ , 1 .
96) ∪ (1 .
96 , ∞ ), we can not reject H
0 at significance level .05.

Test about a Population Mean
1. Parameter of interest: µ = population average strength.
2. Null hypothesis: H
0
: µ = µ
0
= 100.
3. Alternative hypothesis: H a
: µ = 100.
4. Test statistic value: z =
¯ − µ
σ/
√ n
0
=
¯ − 100
2 /
√ n
5. Rejection region: z ≥ z
.
025 or z ≤ −
6. Substituting n = 10 and ¯ = 101 .
10, z
.
025
, where z
.
025
= 1 .
96.
z =
101 .
10 − 100
2 / 10
= 1 .
74
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 13 / 22
7. Since − 1 .
96 < 1 .
74 < 1 .
96, i.e., the value of the test statistic does not fall in the rejection region ( −∞ , 1 .
96) ∪ (1 .
96 , ∞ ), we can not reject H
0 at significance level .05.

Test about a Population Mean
1. Parameter of interest: µ = population average strength.
2. Null hypothesis: H
0
: µ = µ
0
= 100.
3. Alternative hypothesis: H a
: µ = 100.
4. Test statistic value: z =
¯ − µ
σ/
√ n
0
=
¯ − 100
2 /
√ n
5. Rejection region: z ≥ z
.
025 or z ≤ −
6. Substituting n = 10 and ¯ = 101 .
10, z
.
025
, where z
.
025
= 1 .
96.
z =
101 .
10 − 100
2 / 10
= 1 .
74
7. Since − 1 .
96 < 1 .
74 < 1 .
96, i.e., the value of the test statistic does not fall in the rejection region ( −∞ , 1 .
96) ∪ (1 .
96 , ∞ ), we can not reject H
0 at significance level .05.
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 13 / 22

Test about a Population Mean
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 14 / 22
Test for Population Mean of A Normal Population with Known σ
Null hypothesis: H
0
Test statistic value z =
: µ = µ
0
¯ −
σ/
µ
0 n
Alternative Hypothesis
H a
: µ > µ
0
H a
H a
: µ < µ
: µ = µ
0
0
Rejection Region for Level α Test z ≥ z
α
(upper-tailed test) z ≤ − z
α z ≥ z
α/ 2 or z
(lower-tailed test)
≤ − z
α/ 2
(two-tailed test)

Test about a Population Mean
Test for Population Mean of A Normal Population with Known σ
Null hypothesis: H
0
Test statistic value z =
: µ = µ
0
¯ −
σ/
µ
0 n
Alternative Hypothesis
H a
: µ > µ
0
H a
H a
: µ < µ
: µ = µ
0
0
Rejection Region for Level α Test z ≥ z
α z ≤ − z
α
(upper-tailed test)
(lower-tailed test) z ≥ z
α/ 2 or z ≤ − z
α/ 2
(two-tailed test)
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 14 / 22

Test about a Population Mean
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 15 / 22
Example:
To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of 10 welds is selected, and tests are conducted on each weld in the sample. The sample data is recorded as follows
101.9
100.4
101.2
100.9
101.7
101.5
100.9
100.1
101.6
100.8
with X = 101 .
10 .
It is known that the weld strength is normally distributed with mean µ and standard deviation σ = 2 . If the specifications state that the mean strength should be equal to 100 lb/in
2
, shall we accept that the pipe welds meet the specifications with significance level .05? What is the probability of making type II error then?

Test about a Population Mean
Example:
To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of 10 welds is selected, and tests are conducted on each weld in the sample. The sample data is recorded as follows
101.9
100.4
101.2
100.9
101.7
101.5
100.9
100.1
101.6
100.8
with X = 101 .
10 .
It is known that the weld strength is normally distributed with mean µ and standard deviation σ = 2 . If the specifications state that the mean strength should be equal to 100 lb/in
2
, shall we accept that the pipe welds meet the specifications with significance level .05? What is the probability of making type II error then?
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 15 / 22

Test about a Population Mean
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 16 / 22
H
0
: µ = µ
0 v.s.
H a
: µ > µ
0
Then the rejection region for level
¯ ≥ µ
Let µ
0
0
+ z
α
· σ/ n .
α test is z ≥ z
α
, or equivalently denote a particular value of µ that is less than the null value µ
0
, then
β ( µ
0
) = P ( H
0 is not rejected | µ = µ
0
= P ( X < µ
0
+ z
α
· σ/
√ n | µ =
)
µ
0
)
= P
= Φ
X − µ
0
σ/
√ n z
α
+
< z
α
µ
0
σ/
−
√
µ n
0
+
µ
0
σ/
−
√
µ n
0
| µ = µ
0

Test about a Population Mean
H
0
: µ = µ
0 v.s.
H a
: µ > µ
0
Then the rejection region for level
¯ ≥ µ
Let µ
0
0
+ z
α
· σ/ n .
α test is z ≥ z
α
, or equivalently denote a particular value of µ that is less than the null value µ
0
, then
β ( µ
0
) = P ( H
0 is not rejected | µ = µ
0
= P ( X < µ
0
+ z
α
· σ/
√ n | µ =
)
µ
0
)
= P
= Φ z
X − µ
0
σ/
√ n
α
+
< z
α
µ
0
σ/
−
√
µ n
0
+
µ
0
σ/
−
√
µ n
0
| µ = µ
0
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 16 / 22

Test about a Population Mean
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 17 / 22
Alternative Hypothesis Type II Error Probability β ( µ
0
) for Level α Test
H
H
H a a a
:
: µ > µ
µ < µ
: µ = µ
0
0
0
Φ z
α/ 2
Φ z
α
+
1 − Φ − z
α
µ
0
σ/
− µ n
0
+
µ
0
σ/
√ n
0
+
µ
0
σ/
√ n
0
− Φ − z
α/ 2
+
µ
0
σ/
√ n
0

Test about a Population Mean
Alternative Hypothesis Type II Error Probability β ( µ
0
) for Level α Test
H
H
H a a a
:
: µ > µ
µ < µ
: µ = µ
0
0
0
Φ z
α/ 2
Φ z
α
+
1 − Φ − z
α
µ
0
σ/
− µ n
0
+
µ
0
σ/
√ n
0
+
µ
0
σ/
√ n
0
− Φ − z
α/ 2
+
µ
0
σ/
√ n
0
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 17 / 22

Test about a Population Mean
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 18 / 22
Example:
To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of n welds is selected, and tests are conducted on each weld in the sample.
It is known that the weld strength is normally distributed with mean µ and standard deviation σ = 2 . And the specifications state that the mean strength should be equal to 100 lb/in
2
. To construct a hypothesis with
α = .
05 and β = .
1 , how large should n be?

Test about a Population Mean
Example:
To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of n welds is selected, and tests are conducted on each weld in the sample.
It is known that the weld strength is normally distributed with mean µ and standard deviation σ = 2 . And the specifications state that the mean strength should be equal to 100 lb/in
2
. To construct a hypothesis with
α = .
05 and β = .
1 , how large should n be?
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 18 / 22

Test about a Population Mean
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 19 / 22
The sample size n for which a level α test also has β ( µ
0
) = β at the alternative value µ
0 is n =
 h σ ( z
α



+ z
β
µ
0
− µ
0
) i
2



σ ( z
α/ 2
+ z
β
µ
0
− µ
0
)
2 for a one-tailed (upper or lower) test for a two-tailed test (an approximate solution)

Test about a Population Mean
The sample size n for which a level α test also has β ( µ
0
) = β at the alternative value µ
0 is n =
 h σ ( z
α



+ z
β
µ
0
− µ
0
) i
2



σ ( z
α/ 2
+ z
β
µ
0
− µ
0
)
2 for a one-tailed (upper or lower) test for a two-tailed test (an approximate solution)
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 19 / 22

Test about a Population Mean
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 20 / 22
Example:
To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of 50 welds is selected, and tests are conducted on each weld in the sample. The average strength of this sample is X = 101 .
10 , and the standard deviation is s = 2 . If the specifications state that the mean strength should exceed 100 lb/in
2
, shall we accept that the pipe welds meet the specifications with significance level .05?

Test about a Population Mean
Example:
To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of 50 welds is selected, and tests are conducted on each weld in the sample. The average strength of this sample is X = 101 .
10 , and the standard deviation is s = 2 . If the specifications state that the mean strength should exceed 100 lb/in
2
, shall we accept that the pipe welds meet the specifications with significance level .05?
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 20 / 22

Test about a Population Mean
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 21 / 22
Example:
To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of 10 welds is selected, and tests are conducted on each weld in the sample. The sample data is recorded as follows
101.9
100.4
101.2
100.9
101.7
101.5
100.9
100.1
101.6
100.8
with X = 101 .
10 and s = .
585 .
It is known that the weld strength is normally distributed with mean µ . If the specifications state that the mean strength should exceed 100 lb/in
2
, shall we accept that the pipe welds meet the specifications with significance level .05?

Test about a Population Mean
Example:
To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of 10 welds is selected, and tests are conducted on each weld in the sample. The sample data is recorded as follows
101.9
100.4
101.2
100.9
101.7
101.5
100.9
100.1
101.6
100.8
with X = 101 .
10 and s = .
585 .
It is known that the weld strength is normally distributed with mean µ . If the specifications state that the mean strength should exceed 100 lb/in
2
, shall we accept that the pipe welds meet the specifications with significance level .05?
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 21 / 22

Test about a Population Mean
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 22 / 22
Test for Population Mean of A Normal Population with Unknown σ
Null hypothesis: H
0
Test statistic value t =
: µ = µ
0
¯ s /
√ 0 n
Alternative Hypothesis
H a
: µ > µ
0
H a
H a
: µ < µ
: µ = µ
0
0
Rejection Region for Level α Test t ≥ t
α, n − 1
(upper-tailed test) t ≤ − t
α, n − 1 t ≥ t
α/ 2 , n − 1 or t ≤ −
(lower-tailed test) t
α/ 2 , n − 1
(two-tailed test)

Test about a Population Mean
Test for Population Mean of A Normal Population with Unknown σ
Null hypothesis: H
0
Test statistic value t =
: µ = µ
0
¯ s /
√ 0 n
Alternative Hypothesis
H a
: µ > µ
0
H a
H a
: µ < µ
: µ = µ
0
0
Rejection Region for Level α Test t ≥ t
α, n − 1 t ≤ − t
α, n − 1
(upper-tailed test)
(lower-tailed test) t ≥ t
α/ 2 , n − 1 or t ≤ − t
α/ 2 , n − 1
(two-tailed test)
Liang Zhang (UofU)
Applied Statistics I
July 21, 2008 22 / 22