Download Hypothesis Tests

Hypothesis Tests
Hypothesis Tests
• An Hypothesis is a guess about a situation that can be
tested, and the test outcome can be either true or false.
–The Null Hypothesis has a symbol H0, and is always
the default situation that must be proven unlikely
beyond a reasonable doubt.
–The Alternative Hypothesis is denoted by the symbol
HA and can be thought of as the opposite of the Null
Hypothesis - it can also be either true or false, but it is
always false when H0 is true and vice-versa.
Hypothesis Testing Errors
–Type I Errors occur when a test statistic leads us to reject
the Null Hypothesis when the Null Hypothesis is true in
reality.
• The chance of making a Type I Error is estimated by the
parameter  (or level of significance), which quantifies the
reasonable doubt.
–Type II Errors occur when a test statistic leads us to fail to
reject the Null Hypothesis when the Null Hypothesis is
actually false in reality.
• The probability of making a Type II Error is estimated by the
parameter .
Types of Hypothesis Tests
Hypothesis Tests & Rejection Criteria
H0: MA is not different than M0
H0: MA is not better than M0
HA: MA is different than M0
HA: MA is lower than M0

θA θ0
Dm


2
2
θA -θ0
Dm
+θ0 θA
H0: MA is not better than M0
HA: MA is higher than M0

Dm
θ0 θA
One-Sided Test
Statistic < Rejection Criterion
Two-Sided Test
Statistic < -½ Rejection Criterion
or
Statistic > +½ Rejection Criterion
One-Sided Test
Statistic > Rejection Criterion
H0: θA ≥ θ0
HA: θA < θ0
H0: -θ0 ≤ θA ≤ +θ0
HA: θA< -θ0 or +θ0< θA
H0: θA ≤ θ0
HA: θA > θ0
Hypothesis Testing Steps
1. State the null hypothesis (H0) from one of the alternatives:
that the test statistic MA = M0 , MA ≥ M0 , or MA ≤ M0 .
2. Choose the alternative hypothesis (HA) from the alternatives:
MA  M0 , MA < M0 , or MA > M0 . (Respective to above!)
3. Choose a significance level of the test ().
4. Select the appropriate test statistic and establish a critical region.
(If the decision is to be based on a P-value, it is not necessary to have a critical
region)
5. Compute the value of the test statistic () from the sample data.
6. Decision: Reject H0 if the test statistic has a value in the critical
region (or if the computed P-value is less than or equal to the desired
significance level ); otherwise, do not reject H0.
Situation I: Means Test,
Both σ0 and μ0 Known
• Used with:
–an existing process with good deal of data showing the
variation and location are stable
• Procedure:
–use the the z-statistic to compare sample mean with
population mean 0
x  0
z0 =
 0 


 n
Sampling Distribution of the Mean
from the Normal Distribution
• Take a random sample, x1, x2, …, xn, from a normal population
with mean μ and standard deviation σ, i.e., x ~ N(μ, σ )
• Compute the sample average x
• Then x will be normally distributed with mean μ and standard
deviation: σ
n
that is:
 σ 
x ~ N(μ, σ x ) = N μ,

n

Ex. Sampling Distribution of x
• When a process is operating properly, the mean density of a liquid
is 10 with standard deviation 5. Five observations are taken and
the average density is 15.
• What is the distribution of the sample average?
– r.v. x = density of liquid
5 

x ~ N μ = 10, σ =

5

Ex. Sampling Distribution of x
• What is the probability the sample average is greater
than 15?
x  0
15  10
5
Z=
=
=
= 2.23
0
5
2.24
n
5
{ X 15|  the
= 10process
} = P{ Z is2.operating
23} = 0.0129
• Would youPconclude
properly?
Hypothesis Testing
Significance Level of a Hypothesis Test:
A hypothesis test with a significance level or size  rejects the null
hypothesis H0 if a p-value smaller than  is obtained, and accepts the
null hypothesis H0 if a p-value larger than  is obtained. In this case,
the probability of a Type I error (the probability of rejecting the null
hypothesis when it is true) is equal to .
True Situation
Test Conclusion
•
H0 is True
H0 is False
H0 is True
CORRECT
Type II Error ()
H0 is False
Type I Error ()
CORRECT
 error
 = 0.0129
1-
0
2.23
P{ X  15 |  = 10} = 0.0129
In words: If the true mean is 10, the probably that we would see a
sample mean of 15 or higher is only 1.3%. If we say that the mean
has shifted to something larger than 10, there is a 1.3% chance we
are wrong.
 error
Type II error (  ) can only be calculated if we want to test for a specific shift.
Suppose for example that it is important that detect a shift of  = 16 in the liquid
density level.
HO
HA
10
c 12
 = P{ X  c |  =10}
 = P{ X  c |  = 12}
 error
Viscosity is supposed to be 10. I need to be able to detect if viscosity has
shifted to 12. We agree to use 11 as the critical value for the test.
HO
10
HA
c 12
 = P{ X 11|  =10}
= P{ Z 
11  10
} = P{ Z  .45} = 0.326
5
5
 error
Viscosity is supposed to be 10. I need to be able to detect if viscosity has
shifted to 12.
HO
10
HA
c 12
 = P{ X 11|  =12}
= P{ Z 
11  12
} = P{ Z  .45} = 0.326
5
5
 error
Viscosity is supposed to be 10. I need to be able to detect if viscosity has
shifted to 12.
HO
10
HA
 =  = 0.326
c 12
In words: We have a 32.6% chance of buying product when the
mean has shifted. The supplier has a 32.6% chance of taking back
good product. How do we resolve this ?
Situation III: Means Test
Unknown σ(s) and Known μ0
• Used when:
–have good control over the center of the distribution, but the
variation changed from time to time
• Procedure:
–use the the t-statistic to compare both sample means
x  0
t0 =
S
n
v = n – 1 degrees of freedom
Testing Example
• Single Sample, Two-Sided t-Test:
– H0: µ = µ0 versus HA: µ  µ0
– Test Statistic:

x  0 )
T=
s
n
– Critical Region: reject H0 if |T| > t/2,n-1
– P-Value: 2 • P(X  |T|), where the random variable x has a
t-distribution with n _ 1 degrees of freedom
Hypothesis Testing
H0: μ = μ0 versus HA: μ  μ0
tn-1 distribution
-|T|
0
|T|
Statistics and Sampling
• Objective of statistical inference:
– Draw conclusions/make decisions about a population based
on a sample selected from the population
• Random sample – a sample, x1, x2, …, xn , selected so that
observations are independently and identically distributed (iid).
• Statistic – function of the sample data
– Quantities computed from observations in sample and used to
make statistical inferences
– e.g.
measures central tendency
1 n
x =  xi
n i =1
Comparison of Means
–The first types of comparison are those that compare the
location of two distributions. To do this:
• Compare the difference in the mean values for the two
distributions, and check to see if the magnitude of their
difference is sufficiently large relative to the amount of
variation in the distributions
Definitely Different
Probably Different
Probably NOT
Different
Definitely NOT
Different
• Which type of test statistic we use depends on what is known
about the process(es), and how efficient we can be with our
collected data
Situation II: Means Test
σ(s) Known and μ(s) Unknown
• Used when:
–the means from two existing processes may differ, but the
variation of the two processes is stable, so we can estimate
the population variances pretty closely.
• Procedure:
–use the the z-statistic to compare both sample means
z0 =
x1  x 2
 12
n1

 22
n2
Situation IV: Means Test Unknown
σ(s) and μ(s), Similar s2
• Used when:
–logical case for similar variances, but no real "history" with
either process distribution (means & variances)
• Procedure:
–use the the t-statistic to compare using pooled S,
v = n1 + n2 – 2 degrees of freedom
x1  x 2
t0 =
1
1
Sp

n1 n2
(n1  1)S12  (n2  1)S22
Sp =
n1  n2  2
Situation V: Means Test
Unknown σ(s) and μ(s), Dissimilar s2
• Used when:
–worst case data efficiency - no real "history" with either
process distribution (means & variances)
• Procedure:
–use the the t-statistic to compare,
degrees of freedom given by:
t0 =
x1  x2
S12 S22

n1 n2
2
S
S 



n1 n2 

v=
2
2
2
2
 S1 
 S2 
 
 
 n1    n2 
n1  1 n2  1
2
1
2
2
Situation VI: Means Test
Paired but Unknown σ(s)
• Used when:
–exact same sample work piece could be run through both
processes, eliminating material variation
• Procedure:
–define variable (d) for the difference in test value pairs
(di = x1i - x2i) observed on ith sample, v = n - 1 dof
d
t0 =
Sd
n
 d  d)
n
2
i
Sd =
i=1
n 1
Ex. Surface Roughness
• Surface roughness is normally distributed with mean 125
and std dev of 5. The specification is 125 ± 11.65 and we
have calculated that 98% of parts are within specs during
usual production. This has been the case for a long time.
• My supplier of these parts has sent me a large shipment.
I take a random sample of 10 parts. The sample average
roughness is 134 which is within specifications.
• Test the hypothesis that the lot roughness is higher than
specifications at  = 0.05.
e.g. Surface Roughness
Check the hypothesis that the sample of size 10, and with an average of 134
comes from a population with mean 125 and standard deviation of 5.
One-Sided Test
H0:  ≤ 0
HA:  > 0
Test Statistic:
y  0
z0 =

n
Critical Value:
Z = 1.645
Should I reject H0?
z0 =
134  125
9
=
= 5.69
5
1.58
=
10
Alpha One-sided Two-sided
Level (α)
z
z
0.1
1.28155 1.64485
0.05
1.64485 1.95996
Yes! Since 5.69 > 1.645, it is likely that it exceeds the roughness.
e.g. Surface Roughness
• Find the probability that the sample of size 10, and with an
average of 134 does not come from a population with mean
125 and standard deviation of 5.
z0 =
y  0

n
=
134  125
9
z0 =
=
= 5.69
5
1.58
10
P  value = 1   ( z0 ) = 1   (5.69)  1  1 = 0
• Should I accept this shipment?