ST 361: Ch8.1 Hypotheses and Test Procedures
Topics:
► Hypothesis testing
► Statistical hypotheses
► Test procedure
► Errors in hypotheses testing
-------------------------------------------------------------------------------------------------------------------------------I.
Hypothesis Testing
 What is it?
It is a statistical tool that can be used to answer questions like the following:
Ex1. Let  = the average height of NCSU students. From a sample of 100 students we obtain a
point estimate X =5’9”. Assume the average height of 5 years ago is 5’8”.
Q: Has the average height of NCSU students increased?
Ex2. Let  D be the mean lifetime for Duracell batteries, and  E be that of Eveready batteries.
Based on a sample of 90 Duracell batteries and 100 Eveready batteries, we obtain that
X D  X E  0.4
Q: Do Duracell batteries have shorter average lifetime than Eveready batteries?
To answer, we state the question as a certain “hypothesis”, and then test if the hypothesis is
supported by the data.

Statistical Hypothesis = ________________________________________________
So it is a claim about the parameters / statistics (pick one).

Hypothesis Testing = ________________________________________________
________________________________________________
 Components of hypothesis testing
[1] The hypotheses
[2] The test procedure
[3] Possible errors in hypothesis testing
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[1] The hypotheses
Rule (a): Hypotheses must be specified in terms of the __________________
Ex1. Let  = the average height of NCSU students. From a sample of 100 students we
obtain a point estimate X =5’9”. Assume the average height of 5 years ago is 5’8”.
Has the average height of NCSU students increased?
o
Parameter of interest:
o
Claim:
Ex2. Let  D be the mean lifetime for Duracell batteries, and  E be that of Eveready
batteries. Based on a sample of 90 Duracell batteries and 100 Eveready batteries, we
obtain that X D  X E  0.4 . Do Duracell batteries have shorter average lifetime than
Eveready batteries?
o
Parameter of interest:
o
Claim:
Rule (b): Hypothesis must be specified in _________________. We state the question of
interest in terms of two hypotheses:
______________________ vs. _______________________
 Null hypothesis (denoted by H 0 ) : neutral / no difference / no effect
 Alternative hypothesis (denoted by H a ) : a contradictory claim to H 0 . It depends on
the questions we want to answer.
Ex1. Parameter of interest:
o
H0 :
Ha :
Ex2. Parameter of interest:
o
H0 :
Ha :
2
Ex3. Let   mean fat content in hamburger. Fast food chain wants to claim the fat content is
less than 6g.
o
o
Parameter of interest:
H0 :
Ha :
Ex4. Let  be the probability of obtaining a head from tossing a coin. One tossed this coin
100 times and obtained that p  0.4 . Can we conclude that the coin is biased?
o
o
Parameter of interest:
H0 :
Ha :
[2] Test Procedure
1. ____________________________________________________________
2. Quantify the statistical evidence based on ___________________________
Note that evidence is quantified in terms of that assuming ________ is true, how likely one
would obtain data as what observed in the sample data (or more extreme)
►This probability is referred to as __________________________
3. If the evidence does not support H 0 (i.e., ______________________), we reject H 0 and
conclude H a is true. Otherwise, we cannot draw conclusion and therefore continue to
believe that H 0 is true.
p-value
► So we predetermine a threshold for “very small p-value”, and called it ____. Then
► We either “reject H0“ or “fail to reject H0”; we usually do not conclude to “accept H0”
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[3] Error Types in Hypothesis Testing
H 0 is not rejected
H 0 is rejected
H 0 True
H a True

Type I error (  error ) = reject H 0 when H 0 is true
Type II error (  error ) = do not reject when H a is true
Ex1. H 0 :   5'8" and H a :   5'8" .
If the population mean height is 5’9”, but we do not reject H 0 . This is __________ error
Ex2. H 0 : D  E vs. H a : D  E
 Type I Error means:
 Type II Error means:
Ex4. H 0 :   0.5 and H a :   0.5 .
If it is a fair coin and we didn’t reject H 0 . This is ________________________

In practice, one cannot minimize both types of error at the same time, and the two types of
error need to be compromised in a test.
Ex. Image if we set =0:
So should allow for some risk of making type I error (such as setting =0.05), so to gain
ability (power) to reject H0 when appropriate.
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
Meaning of :
► Under H0, p-value can be any values from 0 to 1. So if the threshold of p-value is a, then
Pr( making type I error) = ____________________
cf. Pr(making type II error) is referred to as _______
►  is called _______________________________.
Some Practice Problem:
___________ 1. Which of the following is not a statistical hypothesis?
(a)   100
(b)   100
(c) x  0.5
(d)   100
___________ 2. Which of the following pair of statistical hypotheses is valid?
(a) H 0 :   150 vs. H a :   150
(b) H 0 :   150 vs. H a :   150
(c) H 0 :   120 vs. H a :   100
(d) H 0 :   120 vs. H a :   100
___________ 3. Which of the following statement is true?
(a) The null hypothesis, denoted by H o , is the claim that is initially assumed to be true in the
hypothesis testing.
(b) The alternative hypothesis, denoted by H a , is the assertion that is contradictory to the null
hypothesis H o .
(c) The null hypothesis H o will be rejected in favor of the alternative hypothesis only if sample
evidence suggests that H o is false.
(d) If sample evidence does not strongly contradict the null hypothesis H o , we will continue to
believe in the truth of H o
(e) All of the above statements are true
___________ 4. Let  = P(making a Type I error) and  = P(making a Type II error). The null hypothesis is
rejected when
(a) the p-value is larger than 
(b) the p-value is smaller than 
(c) the p-value is larger than 
(d) the p-value is smaller than 
___________ 5. In hypothesis-testing analysis, a type II error occurs if the null hypothesis H o is
a) Rejected when it is true
b) Rejected when it is false.
c) Not rejected when it is false
d) Not rejected when it is true.
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