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Hypothesis Tests on the Mean
H0:  = 0
 H1:   0

Reject H 0 if
Fail to reject H 0 if
Z0 
X  0

n
Z0  z / 2 or Z0  z / 2
 z / 2  Z0  z / 2
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Hypothesis Tests (one side)
H0:  = 0
Reject H0 if
 H1:  > 0
Z0  z
H0:  = 0
Reject H0 if
 H1:  < 0
Z0   z


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Example: two-sided test
Suppose Guido takes a random sample of
n=25 and obtains an average burn rate of
51.3 cm/s.
 Specs require that burn rate must be 50
cm/s, and the standard deviation is known
to be 2 cm/s.
 He decides to specify a type I error
probability (significance level) of 0.05.
 What conclusions can be drawn?

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P-value

The P-value is the smallest level of
significance that would lead to rejection of
the null hypothesis H0 with the given data.
2 [ 1   (| z0 |)] for test : H 0 :    0 H 1 :    0

P   1   ( z0 )
for test : H 0 :    0 H 1 :    0
 ( z )
for test : H 0 :    0 H 1 :    0
0

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Probability of Type II Error

The probability of the type II error is
the probability that Z0 falls between
-z/2 and z/2 given that H1 is true.


 n
 n
    z / 2 
     z / 2 

 
 


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 Requirement Sample Size
For two - tailed tests
( z / 2  z  ) 
2
n

2
2
,     0
For one - tailes tests
( z  z  ) 
2
n

2
2
,     0
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Example (cont.)

Suppose Guido wants to design the
burn rate test so that if the true mean
burn rate differs from 50 cm/s by as
much as 1 cm/s, the test will detect this
(i.e. reject the null hypothesis) w.p.
0.90. Determine the sample size
required to detect this departure.
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Confidence Intervals
If x is the sample mean of a random sample
of size n from a population with variance  ,
a 100(1   )% CI on  is given by
2

 

, x  z / 2
 x  z / 2

n
n

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Example
Construct a 95% confidence interval for
the burn rate. 25 samples were taken.
The sample mean is 51.3 and the
standard deviation is 2.
 What is the relationship between
hypothesis testing and confidence
intervals?

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Error Bound Sample Size
If x is an estimate of  , we can be 100(1   )%
confident that the error |x   | will not exceed a
specified amount E when the sample size is:
 z / 2 
n

 E 
2
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Example

Suppose that we wanted the error in
estimating the mean burn rate of the
rocket propellant to less than 1.5 cm/s,
with 95% confidence. What is the
required sample size?
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One-Sided CIs
The 100(1   )% upper-CI for  is
  x  z

n
The 100(1   )% lower-CI for  is
  x  z

n
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General CIs
Let ˆ be an estimator for  .
The 100(1   )% CI, (L,U ), is given by:


P L  ˆ  U  1  
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