3. Hypotheses
The null hypothesis, denoted H0, is the
claim that is initially assumed to be true.
The alternative hypothesis, denoted by
Ha, is the assertion that is contrary to H0.
Possible conclusions from hypothesis-
testing analysis are reject H0 or fail to
reject H0.
4. Hypotheses
H0 may usually be considered the
skeptic’s hypothesis: Nothing new or
interesting happening here! (And
anything “interesting” observed is due to
chance alone.)
Ha may usually be considered the
researcher’s hypothesis.
5. Rules for Hypotheses
H0 is always stated as an equality claim
involving parameters.
Ha is an inequality claim that contradicts
H0. It may be one-sided (using either >
or <) or two-sided (using ≠).
6. A Test of Hypotheses
A test of hypotheses is a method for
using sample data to decide whether
the null hypothesis should be
rejected.
7. Test Procedure
A test procedure is specified by
• A test statistic, a function of the
sample data on which the decision is
to be based.
• (Sometimes, not always!) A
rejection region, the set of all test
statistic values for which H0 will be
rejected
8. Errors in Hypothesis Testing
A type I error consists of rejecting the
null hypothesis H0 when it was true.
A type II error consists of not rejecting
H0 when H0 is false.
α and β are the probabilities of type
I and type II error, respectively.
9. Level α Test
Sometimes, the experimenter will fix
the value of α , also known as the
significance level.
A test corresponding to the significance
level is called a level α test. A test
with significance level α is one for
which the type I error probability is
controlled at the specified level.
10. Rejection Region: α and β
Suppose an experiment and a sample
size are fixed, and a test statistic is
chosen. 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.
12. Case I: A Normal Population
With Known σ
Null hypothesis: H 0 : µ = µ0
x − µ0
Test statistic value: z =
σ/ n
13. Case I: A Normal Population
With Known σ
Alternative Rejection Region
Hypothesis for Level α Test
H a : µ > µ0 z ≥ zα
H a : µ < µ0 z ≤ − zα
H a : µ ≠ µ0 z ≥ zα / 2 or z ≤ − zα / 2
14. Recommended Steps in
Hypothesis-Testing Analysis
1. Identify the parameter of interest and
describe it in the context of the
problem situation.
2. Determine the null value and state
the null hypothesis.
3. State the alternative hypothesis.
15. Hypothesis-Testing Analysis
1. Give the formula for the computed
value of the test statistic.
2. State the rejection region for the
selected significance level
3. Compute any necessary sample
quantities, substitute into the formula
for the test statistic value, and
compute that value.
16. Hypothesis-Testing Analysis
7. Decide whether H0 should be
rejected and state this conclusion in
the problem context.
The formulation of hypotheses (steps 2
and 3) should be done before examining
the data.
17. Type II Probability β ( µ ′) for a Level α
Test Type II
Alt. Hypothesis Probability β ( µ ′)
µ0 − µ ′
H a : µ > µ0 Φ zα +
σ/ n
µ0 − µ ′
H a : µ < µ0 1 − Φ − zα +
σ/ n
H a : µ ≠ µ0
Φ zα / 2 +
µ0 − µ ′
− Φ − zα / 2 +
µ0 − µ ′
σ/ n σ/ n
18. Sample Size
The sample size n for which a level α
test also has β ( µ ′) = β at the alternative
value µ ′ is
σ ( zα + z β ) 2
one-tailed test
µ0 − µ ′
n= 2
σ ( zα / 2 + z β )
two-tailed test
µ0 − µ ′
19. Case II: Large-Sample Tests
When the sample size is large, the z
tests for case I are modified to yield
valid test procedures without
requiring either a normal population
distribution or a known σ .
20. Large Sample Tests (n > 40)
For large n, s is close to σ .
X − µ0
Test Statistic: Z =
S/ n
The use of rejection regions for case I
results in a test procedure for which the
significance level is approximately α .
21. Case III: A Normal Population
Distribution
If X1,…,Xn is a random sample from a
normal distribution, the standardized
variable X −µ
T=
S/ n
has a t distribution with n – 1
degrees of freedom.
22. The One-Sample t Test
Null hypothesis: H 0 : µ = µ0
x − µ0
Test statistic value: t=
s/ n
23. The One-Sample t Test
Alternative Rejection Region
Hypothesis for Level α Test
H a : µ > µ0 t ≥ tα ,n −1
H a : µ < µ0 t ≤ −tα ,n −1
H a : µ ≠ µ0 t ≥ tα / 2,n −1 or t ≤ −tα / 2,n −1
24. A Typical β Curve for the t Test
β curve for n – 1 df
β when
µ = µ′
0
Value of d corresponding to
specified alternative to µ ′
29. Large-Samples Concerning p
Alternative Rejection Region
Hypothesis
H a : p > p0 z ≥ zα
H a : p < p0 z ≤ − zα
H a : p ≠ p0 z ≥ zα / 2 or z ≤ − zα / 2
Valid provided
np0 ≥ 10 and n(1 − p0 ) ≥ 10.
30. General Expressions for β ( p′)
Alt. Hypothesis β ( p′)
H a : p > p0 p0 − p′ + zα p0 (1 − p0 ) / n
Φ
p′(1 − p′) / n
p0 − p′ − zα p0 (1 − p0 ) / n
H a : p < p0 1− Φ
p′(1 − p′) / n
31. General Expressions for β ( p′)
Alt. Hypothesis β ( p′)
p0 − p′ + zα p0 (1 − p0 ) / n
H a : p ≠ p0 Φ
p′(1 − p′) / n
p0 − p′ − zα p0 (1 − p0 ) / n
−Φ
p′(1 − p′) / n
32. Sample Size
The sample size n for which a level α
test also has β ( p′) = p
z p (1 − p ) + z p′(1 − p′) 2 one-tailed
α 0 0 β
p′ − p0 test
n= 2
zα / 2 p0 (1 − p0 ) + z β p′(1 − p′) two-tailed
p′ − p0 test
33. Small-Sample Tests
Test procedures when the sample size
n is small are based directly on the
binomial distribution rather than the
normal approximation.
P (type I) = 1 − B (c − 1; n, p0 )
B ( p′) = B (c − 1; n, p′)
35. P - Value
The P-value is the smallest level of
significance at which H0 would be
rejected when a specified test procedure
is used on a given data set.
1. P-value ≤ α
⇒ reject H 0 at a level of α
2. P -value > α
⇒ do not reject H 0 at a level of α
36. P - Value
The P-value is the probability,
calculated assuming H0 is true, of
obtaining a test statistic value at least as
contradictory to H0 as the value that
actually resulted. The smaller the P-
value, the more contradictory is the data
to H0.
37. P-Values for a z Test
P-value:
1 − Φ( z ) upper-tailed test
P= Φ( z ) lower-tailed test
2 1 − Φ ( z ) two-tailed test
38. P-Value (area)
P-value = 1 − Φ ( z )
Upper-Tailed
P -value = Φ ( z ) 0 z
Lower-Tailed
-z 0
P-value = 2[1 − Φ (| z |)] Two-Tailed
-z 0 z
39. P–Values for t Tests
The P-value for a t test will be a t
curve area. The number of df for the
one-sample t test is n – 1.
41. Constructing a Test Procedure
1. Specify a test statistic.
2. Decide on the general form of the
rejection region.
3. Select the specific numerical critical
value or values that will separate the
rejection region from the acceptance
region.
42. Issues to be Considered
1. What are the practical implications
and consequences of choosing a
particular level of significance once
the other aspects of a test procedure
have been determined?
2. Does there exist a general principle
that can be used to obtain best or good
test procedures?
43. Issues to be Considered
1. When there exist two or more tests that
are appropriate in a given situation, how
can the tests be compared to decide
which should be used?
2. If a test is derived under specific
assumptions about the distribution of
the population being sampled, how well
will the test procedure work when the
assumptions are violated?
44. Statistical Versus Practical
Significance
Be careful in interpreting evidence when
the sample size is large, since any small
departure from H0 will almost surely be
detected by a test (statistical significance),
yet such a departure may have little
practical significance.
45. The Likelihood Ratio Principle
1. Find the largest value of the likelihood
for any θ in Ω0 .
2. Find the largest value of the likelihood
for any θ in Ωa .
3. Form the ratio
maximum likelihood for θ in Ω0
λ ( x1 ,..., xn ) =
maximum likelihood for θ in Ωa
Reject H0 when this ratio is small.
