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AMS7: WEEK 6. CLASS 2
Hypothesis Testing with One Sample
Friday May 08, 2015
Test Statistic
• Value computed from the data and used to make the
decision about the rejection of the Null Hypothesis
PARAMETER
TEST STATISTIC
Population Proportion (p)
Population Mean ():
known or unknown
Population Standard
Deviation ()
=
=
௑തିఓ
഑
೙
or
̂ − . t=
௑തିఓ
ೞ
೙
ଶ
−
1
ଶ =
ଶ
Hypothesis Testing (Cont.)
• We assume Ho is true and use the test statistic for
determining whether there is significant evidence against
Ho.
Critical Region (or Rejection Region): Set of all values of
the test statistic that cause us to Reject Ho.
Significance level (): Probability that the test statistic will
fall in the critical region when Ho is true.
ߙ is the probability of making the mistake of
rejecting Ho when Ho is true
Hypothesis Testing (Cont.)
• Values of ߙ: 0.05, 0.01 and 0.10. The most commonly
used value is 0.05 (Default value)
• Two-tailed test: Critical region is in two tails
• Left-tailed test: Critical region is in the left tail
• Right-tailed test: Critical region is in the right tail
• Critical Value: Separates the critical Region (where we
reject Ho) from the values of the test statistic that do not
lead to rejection of the Ho Hypothesis.
Types of Tests (Assume
Two-tailed Test
/2=0.025
/2=0.025
z=-1.96 z=0
Critical Region
z=1.96
Critical Region
Types of Tests (Assume
Left-tailed Test
=0.05
z=-1.645 z=0
Critical Region
Types of Tests (Assume
Right-tailed Test
=0.05
z=0
z=1.645
Critical Region
How do we know what kind of test to use?
We need to examine H1’s sign:
• ≠ corresponds to two-tailed tests
• < corresponds to left-tailed tests
• > corresponds to right-tailed tests
Examples: Claim or hypothesis in
symbolic form
1) More than one-half of all internet users make on-line
purchases
Ho: p≤ 0.5
Ho: p= 0.5
H1: p>0.5 (original claim)
H1: p>0.5
Right-tailed test
2) The percentage of viewers tuned to 60 Minutes is equal
to 24%
Ho: p=0.24 (original claim)
H1: p≠0.24
Two-tailed test
Examples (Cont.)
3) Mean IQ of Statistic students is at least 110
Ho: ≥110 (original claim)
Ho: =110
H1: <110
H1: <110
Left-tailed test
Notes:
-Make Ho the hypothesis with the equal sign and H1 the
hypothesis which does not contain an equality.
- For practical reasons Ho is converted to an equal sign
hypothesis.
Examples of Critical Values
Example 1
=0.10
=0.10
z=0
zߙ=1.285
Critical Region
Right-tailed Test
Examples of Critical Values (Cont.)
Example 2 =0.20
/2=0.10
/2=0.10
zߙ/2 =-1.285 z=0
zߙ/2 =1.285
Critical Region
Critical Region
Two-tailed Test
Examples of Critical Values (Cont.)
Example 3 =0.05
=0.05
zߙ =-1.645 z=0
Critical Region
Left-tailed Test
Example: Test claim that more than half of
all Internet users make on-line purchases
• Ho: p= 0.5
H1: p>0.5 (Original Claim)
• Suppose that in a sample of n=1025 subjects, 69% said
they used Internet for shopping (Binomial random
variable. Check np≥5, nq ≥5 for using Normal
approximation)
• Test Statistic: =
௣ොି௣
೛.೜
೙
• Assume Ho is true. This implies that p=0.5 and q=1-
p=0.5. ̂ =0.69 (from the sample!).
Example (Cont.)
•=
଴.଺ଽି଴.ହ
బ.ఱ⨯బ.ఱ
భబమఱ
=
଴.ଵଽ
଴.଴ଵହ଺
= 12.179
• Decision: Test statistic falls in the Critical Region
Reject Ho
=0.10
z=0
zߙ=1.285
Critical Region
12.179 falls in
the critical region
Example (Cont.)
• Interpretation of the Result
We reject the Null hypothesis Ho. We support the
alternative hypothesis H1 which is the original claim.
Sample data provide enough evidence to support the claim
that more than 50% of internet users make on-line
purchases.
Using the Confidence Interval for
Hypothesis Testing
• Confidence Interval for a Proportion
CI: − < < + = ఈൗ ×
ଶ
In this example: =0.10, = 0.69; = 1 − 0.69 = 0.31
Confidence level: (1-2⨯)100%= 80%
E= ଴.ଵ଴ ×
଴.଺ଽ×଴.ଷଵ
ଵ଴ଶହ
= 0.0186.
0.69 − 0.0186 < < 0.69 + 0.0186
Confidence Interval: 0.6714<p<0.7086
Conclusion: p=0.5 is not included in the CI: Reject Ho
Types of Errors
TRUE STATE OF NATURE
DECISION
Ho is TRUE
Ho is FALSE
REJECT Ho
TYPE I ERROR
OK
DO NOT REJECT
Ho
OK
TYPE II ERROR
Types of Errors
• Type I Error: Rejecting Ho when Ho is TRUE. is the
Probability of Type I error (P(Reject (Ho| Ho is True))
• Type II Error: Do not reject Ho when Ho is False. The
Greek letter (beta) is the probability of a type II error
(P(Do not reject Ho| Ho is false))
Note: ߚ≠1-ߙ. Normally we select ߙ first!
For a given sample size n, a decrease in alpha () will
cause and increase in , and conversely, an increase in will cause a decrease in .
Power of a test: Is the probability 1- = Probability of Reject
Ho when is false
The p-value: Another way to take a
decision
• The p-value: Probability of getting a value of the test
statistic greater (in absolute value) or equal to the sample
data value.
EXAMPLE:
= 0.10
z*= Observed Test
Statistic
Zߙ= Critical Value
Right-tailed test:
-If p-value is greater than
do not reject Ho
- If p-value is lower than
reject Ho
p-value:
Area to the right of z*
= 0.10
Area to the right
of Zߙ
Z*
Zߙ=1.285
Example of calculating a p-value
• Suppose H1: p>0.29 and the observed test statistic is
z=1.97. Assume =0.01
• P-value: 1-0.9756=0.0244
Right-tailed test
P-value is this area
P-value > 0.01
Do not Reject Ho
Z=1.97
More on p-value
• Right-tailed test: p-value is the area to the right of the test
statistic z
• Left-tailed test: p-value is the area to the left of the test
statistic z
• Two-tailed test: p-value is twice the area to the extreme
region bound by the test statistics z.
Summary about Hypothesis Testing
Test a claim about a Population
Parameter
Hypothesis test Procedure
Population Proportion (p)
1) Check that the binomial distribution
of sample proportions can be
approximated by a normal
distribution (np≥5; nq≥5)
௣ොି௣
2) Use the test statistic: = ೛.೜
೙
Population mean ( is known)
1) Check that population is normally
distributed or n>30.
2) Use the test statistic: =
Population mean ( is unknown)
௑തିఓ
഑
೙
1)Check that population is normally
distributed or n>30. Use s instead of .
2) Use the test statistic: t =
௑തିఓ
ೞ
೙