Download Chapter 7 Hypothesis Testing

Section 7-1 & 7-2
Overview and Basics of
Hypothesis Testing
Created by Erin Hodgess, Houston, Texas
Chapter 7
Hypothesis Testing
7-1
Overview
7-2
Basics of Hypothesis Testing
7-4
Testing a Claim About a Mean:  Known
7-5
Testing a Claim About a Mean:  Not Known
7-3
Testing a Claim About a Proportion
7.2 Fundamentals of Hypothesis
Testing
Recall: Inferential Statistics – draw conclusions
about a population based on sample data:
Confidence Intervals: Estimate the value of a
population parameter (using sample statistics).
Hypothesis Tests:Tests a claim (hypothesis)
about a population parameter (using sample
statistics).
Basic Idea for Hypothesis Tests
State the hypothesis.
Gather sample data (“evidence”).
Make a decision about your hypothesis,
based on sample data.
Rare Event Rule for
Inferential Statistics
If, under a given assumption, the
probability of a particular observed event
is exceptionally small, we conclude that
the assumption is probably not correct.
Components of a Hypothesis Test
A.
B.
C.
D.
The Null and Alternative Hypotheses
The test statistic and making decisions
Types of errors in hypothesis tests
Writing conclusions
A. The Hypotheses
 Null Hypothesis, H0 :
 statement about the parameter (μ, p, σ)
 assumed true until proven otherwise
 must contain equality sign:
 or  or =
 Alternative Hypothesis, H1 or Ha :
 statement about the parameter that must be true if
H0 is false.
 must contain:
> or < or 
The Hypotheses
Symbolic Form of the Hypotheses
H 0 :   0
H a :   0
H 0 :   0
H 0 :   0
or
or
H a :   0
H a :   0
Note: The original claim* may be in the
null or the alternative hypothesis.
B. Making Decisions
1. State Hypotheses. **Assume H0 is true.
2. Collect sample data (test statistic)
3. **Make a decision about H0:
a)Reject H0 , or
b)Fail to Reject the H0
Note: a) Reject Ha → Accept Ha
b) Fail to Reject the H0 → Can’t Accept Ha
C. Errors in Decision Making
 Type I error is the mistake of rejecting the null
when it is true:
Reject Ho | Ho is true
A Type II error is the mistake of failing to reject
the null when it is false:
FTR Ho | Ho is false
 The symbol (alpha) is used to represent the probability
of a type I error. P(Type I error) = 
 The symbol  (beta) is used to represent the probability
of a type II error. P(Type II error) = 
D. Writing Conclusions
All conclusions are written in terms of the original claim.
How the conclusion is worded depends on:
A.
B.
1.
Whether the original claim was in the null or the alternative, and
2.
Whether the decision was Reject Ho (which implies you accept
the Ha) or FTR Ho (which implies you cannot accept the Ha).
Ho contains the claim, and you:
1.
Reject Ho: “There’s enough evidence to reject the claim that …..
2.
FTR Ho: “There is not enough evidence to reject the claim that…”
Ha contains the claim, and you:
1.
Reject Ho: “There is enough evidence to support the claim that….”
2.
FTR Ho: “There is not enough evidence to support the claim that…”
Wording of Final Conclusion
Figure 7-7
Note about Forming Your
Own Claims (Hypotheses)
If you are conducting a study and want
to use a hypothesis test to support your
claim, the claim must be worded so that
it becomes the alternative hypothesis.
How to Make a Decision
about the Ho
We start with tests about the population
mean, μ.
x
1. State Ho, Ha. **Assume H0 is true: μ = μ0
2. Collect sample data. Compute the test
statistic. (For tests about μ, use x ).
3. Recall: For x~N or n>30 and σ known:
x

N ,
n

How to Make a Decision
about the Ho
4. Determine the Tails of the Test:
a) Left-tailed Test
x
b) Right-Tailed Test
c) Two-Tailed Test
Basic Idea: Assuming the Ho is true, (i.e, μ =
μ0 ) where would the sample mean have to
fall to convince you to reject the Ho and
accept the Ha?
Left-tailed Test
H0: μ ≥ μ0
Ha: μ < μ0
Points Left
Right-tailed Test
H0: μ ≤ μ0
Ha: μ > μ0
Points Right
Two-tailed Test
H0: μ = μ0
Ha: μ ≠ μ0
Means less than or greater than
How to Make a Decision
about the Ho
5. Once you have determined the tails of
the test, use your sample mean to
make a decision about the Ho using
either:
a) The Traditional Method
b) *The P-Value Method
Main Idea**: How unusual would your
sample mean have to be for you to
reject the Ho and accept the Ha??
The Traditional Method
 Critical Region: contains values of the test
statistic that would be considered unusual,
assuming the Ho is true.
 If the test statistic falls in the critical region,
that would be considered unusual, assuming
the Ho was true. DECISION: REJECT HO.
 If the test statistic falls in the non-critical
regions, then that would not be considered
unusual. DECISION: FTR Ho.
Finding the Critical
Region
 Critical Region: contains values of the test
statistic that would be considered unusual,
assuming the Ho is true.
 Area of the Critical Region = α (α is also
known as the significance level of the test).
It is also the probability of a Type I error.
 Critical Values: values that mark the
boundaries of the critical region, zα
Section 7-4
Testing a Claim About a
Mean:  Known
Created by Erin Hodgess, Houston, Texas
Assumptions for Testing
Claims About
Population Means
1) The sample is a simple random
sample.
2) The value of the population standard
deviation  is known.
3) Either or both of these conditions is
satisfied: The population is normally
distributed or n > 30.
H.T about a Mean (with  known)
1. Use the claim to write Ho, Ha.
**Assume the Ho is true.
2. Determine whether you’re doing a lefttailed, right-tailed, or 2-tailed test.
3. Note α, the significance level of the
test.

4. Get sample mean, x. x N o ,  n
Sketch the sampling distribution.

H.T about a Mean (with  Known)
4. Convert the sample mean to a z-score:
x 
z
 n
5. Make a decision about Ho using
either:
a. Traditional Method
b. P-value Method
6. Write a conclusion in terms of the
original claim.
Traditional Method
1. Shade the critical region
2. Find the critical values
3. Make a decision:
a. If z is in the critical region,
reject Ho.
b. If z is not in the critical
region, FTR the Ho.
P-Value Method for Making
Decisions
Instead of determining whether your
sample mean is unusual just by
whether or not it falls in a “critical
region”, be more precise:
“Find
the “P-value”
P-Value Method for Making
Decisions
P-value = probability of getting a test
statistic as extreme or more as the one
from the sample data (assuming the Ho
is true to begin with).
If P-value ≤ α (unusual), Reject Ho.
If Pvalue > α (not unusual), FTR HO.
Example: Finding P-values.
Figure 7-6
Section 7-5
Testing a Claim About a
Mean:  Not Known
Created by Erin Hodgess, Houston, Texas
H.T about Mean (with  unknown)
1. Follow the same procedure as 7.3.
2. For population ≈ normal or n > 30 and
σ unknown, the sampling distribution
of x has a t-distribution with n-1
degrees of freedom.
3. Convert sample mean x to a standard
t-score:
x 
t
s
n
Choosing between the
Normal and Student t
Distributions when Testing a Claim
about a Population Mean µ
Use the Student t distribution when  is not
known and either or both of these
conditions is satisfied: The population is
normally distributed or n > 30.
Section 7-3
Testing a Claim About a
Proportion
Created by Erin Hodgess, Houston, Texas
Hypotheses About p
H 0 : p  p0
H a : p  p0
H 0 : p  p0
H 0 : p  p0
or
or
H a : p  p0
H a : p  p0
Note: The original claim* may be in the
null or the alternative hypothesis.
Assumptions about a
Hypotheses Test for a
Proportion
1. To conduct a hypothesis test about a
population proportion p, use a sample
proportion, p.
2. We assume the sample was a simple
random sample.
3. If np ≥ 5, nq ≥ 5, the sampling distribution of
p is:

pq 
pˆ
N  p,


n 
4. Follow the procedures of 7.3:
Hypothesis Test about a
Population Proportion
1. Use the claim to write Ho, Ha.
**Assume the Ho is true.
2. Determine whether you’re doing a lefttailed, right-tailed, or 2-tailed test.
3. Note α, the significance level of the
test.
4. Get sample proportion, p.
Sketch the sampling distribution.
Hypothesis Test about a
Proportion
4. Convert the sample proportion to a zp̂  p
score:
z
pq
n
5. Make a decision about Ho using
either:
a. Traditional Method
b. P-value Method
6. Write a conclusion in terms of the
original claim.
CAUTION

 When the calculation of p results in a
decimal with many places, store the
number on your calculator and use all
the decimals when evaluating the z test
statistic.

 Large errors can result from rounding p
too much.
Section 7-6
Testing a Claim About a
Standard Deviation or
Variance
Created by Erin Hodgess, Houston, Texas
Assumptions for Testing
Claims About  or 2
1. The sample is a simple random
sample.
2) The population has values that are
normally distributed (a
strict requirement).
Chi-Square Distribution
Test Statistic
2=
(n – 1) s 2
2
Chi-Square Distribution
Test Statistic
2=
n
(n – 1) s 2
2
= sample size
s 2 = sample variance
2 = population variance
(given in null hypothesis)
P-values and Critical
Values for
Chi-Square Distribution
Use Table A-4.
The degrees of freedom = n –1.
Properties of Chi-Square
Distribution
All values of 2 are nonnegative, and the
distribution is not symmetric (see Figure
7-12).
There is a different distribution for each
number of degrees of freedom (see
Figure 7-13).
The critical values are found in Table A-4
using n – 1 degrees of freedom.
Properties of Chi-Square
Distribution
Properties of the Chi-Square
Distribution
Chi-Square Distribution for 10
and 20 Degrees of Freedom
There is a different distribution for each
number of degrees of freedom.
Figure 7-12
Figure 7-13
Example: For a simple random sample of adults, IQ scores are
normally distributed with a mean of 100 and a standard deviation of 15.
A simple random sample of 13 statistics professors yields a standard
deviation of s = 7.2. Assume that IQ scores of statistics professors are
normally distributed and use a 0.05 significance level to test the claim
that  = 15.
H0:  = 15
H1:   15
 = 0.05
n = 13
s = 7.2
 =
2
(n – 1) s 2
2
2
(13
–
1)(7.2)
=
= 2.765
152
Example: For a simple random sample of adults, IQ scores are
normally distributed with a mean of 100 and a standard deviation of 15.
A simple random sample of 13 statistics professors yields a standard
deviation of s = 7.2. Assume that IQ scores of statistics professors are
normally distributed and use a 0.05 significance level to test the claim
that  = 15.
H0:  = 15
H1:   15
 = 0.05
n = 13
s = 7.2
2 = 2.765
Example: For a simple random sample of adults, IQ scores are
normally distributed with a mean of 100 and a standard deviation of 15.
A simple random sample of 13 statistics professors yields a standard
deviation of s = 7.2. Assume that IQ scores of statistics professors are
normally distributed and use a 0.05 significance level to test the claim
that  = 15.
H0:  = 15
H1:   15
 = 0.05
n = 13
s = 7.2
2 = 2.765
The critical values of 4.404 and 23.337 are
found in Table A-4, in the 12th row (degrees
of freedom = n – 1) in the column
corresponding to 0.975 and 0.025.
Example: For a simple random sample of adults, IQ scores are
normally distributed with a mean of 100 and a standard deviation of 15.
A simple random sample of 13 statistics professors yields a standard
deviation of s = 7.2. Assume that IQ scores of statistics professors are
normally distributed and use a 0.05 significance level to test the claim
that  = 15.
H0:  = 15
H1:   15
 = 0.05
n = 13
s = 7.2
2 = 2.765
Because the test statistic is in the critical
region, we reject the null hypothesis. There
is sufficient evidence to warrant rejection of
the claim that the standard deviation is equal
to 15.