DISTRIBUTIONS AND HYPOTHESIS TESTING
Area Under the Normal Curve
35.0  2.5
30.4
(-3)
32.6
(-2)
34.8
(-)
37
()
39.2
(+)
43.6
41.4 (+3)
(+2)
The Normal curve is symmetric about the mean, so the area under the curve may be determined from a
table that has entries for only half of the values. The Standard Normal Distribution has a mean of 0 and a
standard deviation of 1; and the total area under the curve is exactly 1. In order to adapt the table to a
generic situation, we compute a change in variable, defined as:
z
x

where:
x is the point in the interval,
 is the distribution mean, and
 is the distribution standard deviation.
When we do this, z now measures how many standard deviation units the x-value is from the mean. The
table lists the area [(z)] under the normal curve from - to z, so the entries in the table require us to look
up the value of  corresponding to the computed z. [Note: Due to symmetry, (-z) = 1 - (z)]
Example:
A process has a distribution that is Normally distributed, with a mean of 37.0 and a standard deviation of
2.2. The specification calls for a value of 35.0  2.5. Estimate the proportion of the process output that
will meet specifications.
Solution:
1. Convert the problem to a Standard Normal Distribution and find the area under the Standard
Normal Curve from - to the upper specification limit.
2. Convert the problem to a Standard Normal Distribution and find the area under the Standard
Normal Curve from - to the lower specification limit
3. Subtract (2.) from (1.) to get the process yield.
Step 1.)
 = 37
 = 2.2
x  35.0  2.5  37.5
37.5  37.0
z
 0.23
2.2
(0.23)  0.5910
Step 2.)
 = 37
 = 2.2
x  35.0  2.5  32.5
32.5  37.0
z
 2.05
2.2
(  2.05)  1  ( 2.05)  1  0.9798  0.0202
Step 3.) 0.5910  0.0202  0.5708  57%
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Hypothesis Testing Terms
Statistic is an estimate of a parameter or calculation about or from a distribution. A distribution is a
mathematical function that captures the frequency (or probability) that a given situation will occur.
Population is a term that refers to the complete set of observations. If we have the complete set (or even
a very large set of observations), then our estimates (statistics) should be very accurate. We have
special identifiers for these estimates:  is the symbol for the population mean; 2 is the symbol for the
population variance (  is for the population standard deviation); and N will represent the population
size, or the number of elements in the population.
Sample is a term that refers to a subset of the population, selected at random (meaning every element
has an equal chance of being selected), and selected independently (meaning any of our prior selections
did not bias our current selection). We also have special identifiers for the statistics estimated from
samples: x is the symbol for the sample mean; S2 is the symbol for the sample variance (S is the
symbol for the sample standard deviation), and n is the sample size.
Hypothesis is a guess about a situation, that can be tested and can be either true or false. The Null
Hypothesis has a symbol H0, and is always the default situation that must be proven beyond a
reasonable doubt. The Alternative Hypothesis is denoted by the symbol HA (H1 in some texts) 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. There are two types of comparisons, and three forms for the
alternative hypothesis, illustrated in Figure 1 (below).
The first type of HA says that there is a difference, but the direction of the difference is not known (or
needed). In this case, it is a two-sided comparison, and the Null hypothesis is rejected if the test
statistic would be placed in either of the tail regions representing 1/2 of the rejection criteria area (see Fig.
1(b)). Otherwise, the Null hypothesis is kept.
The second type of HA says that there is a difference, and the direction of the difference is anticipated (or
required). In this case, there is a one-sided comparison, and the Null hypothesis is rejected if the test
statistic would be placed in the tail region representing the entire rejection criteria area. If a sample mean
must be greater than the other mean (Form 1), the right tail region is used (Fig. 1(c)), and the value of
the test statistic must exceed the criterion value. If the situation is reversed, and the sample mean must
be less than the other mean (Form 2), then the left tail region is used (Fig. 1(a)) and the criterion value
must be less than the test statistic value. Only one of these last two forms is hypothesized and tested otherwise the two-sided comparison must be used.
Figure 1. Hypothesis Test Rejection Criteria

2

s
0
One-Sided Test
Statistic < Rejection Criterion
(a)
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
2
s
0

0
s
Two-Sided Test
Statistic < -½ Rejection Criterion
or
Statistic > +½ Rejection Criterion
(b)
s
One-Sided Test
Statistic > Rejection Criterion
(c)
Page 2 of 8
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 . Figure 2 depicts the errors associated with
hypothesis testing. Depending on the situation, we can knowingly adjust our tests so that we can control
the risks of these types of errors to a reasonable extent.
H0 is the Null Hypothesis (always the default), HA is the Alternative Hypothesis
H0: There is NO significant difference at the  level.
HA: There IS a significant difference at the  level.
Common  levels: .10, .05, .01
P-value: The smallest level of
significance that would lead
one to reject H0. It is
possible to report the
P-value for the test, and let
the user decide if the
significance is acceptable
(rather than use a set value).
H0 is True
H0 is False
H0 is
True
OK
TYPE II ERROR
 - risk
(Keep H0 when H0 is False)
Consumer's Risk
H0 is
False
TYPE I ERROR
 - risk
(Reject H0 when H0 is True)
Producer's Risk
OK
T
E
S
T
R
E
S
U
L
REALITY
T
Figure 2. Hypothesis Testing & Error Types
Sampling Distributions
It can be difficult to associate an exact distribution with every possible type of event that we might be
interested in predicting. Fortunately, most of the comparison hypotheses that we are interested in making
can be fit to about four, well-known functions. These functions are referred to as sampling
distributions: (1) the Normal Distribution; (2) the t–Distribution; (3) the Chi-Squared (2)
Distribution; and (4) the F–Distribution. One of the factors that controls how well these sampling
distributions approximate the actual distribution depends on the sample size or a statistic calculated from
the sample size called the degrees of freedom (symbol ).
Comparison of Means
One of the first types of comparison that are important are those that compare the location of two
distributions. When we do this, we 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. This is illustrated in Figure 3, below.
Definitely Different
Probably Different
Probably NOT Different
Figure 3. Comparison of Means, Illustrated.
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Definitely NOT Different
Remembering that there will be a little variation in any sample taken from a distribution, the idea when
comparing means it to express the distance between the two means in terms of the spread, and then see
how likely we would be to observe a difference of that magnitude if there really were no difference in the
two distributions (the Null Hypothesis). If the probability of that magnitude is sufficiently small, then we
would reject the Null Hypothesis, and accept the Alternative Hypothesis instead.
Assuming that the mean value from both distributions is normally distributed, we could always use a form
of the t-test to perform the comparison. However, if we know a little bit more about the distributions, we
can be far more efficient with the data, and more effective in detecting a true difference. Figure 5 shows
a decision chart for two-way comparisons. The top portion of the chart shows the different ways that we
could perform the test, based on our knowledge or prior experience with the processes.
Comparison of Variances
A second type of important comparison is when a difference in the spread of two distributions is
suspected. To make this comparison, we compute the ratio of the two variances, and then compare the
ratio to one of two known distributions as a check to see if the magnitude of that ratio is sufficiently
unlikely for the distribution. This is illustrated in Figure 4, below.
Definitely Different
Probably Different
Probably NOT Different
Definitely NOT Different
Figure 4. Comparison of Variances, Illustrated.
Again, similar to our comparison of means, more experience with at least one of the processes will let us
be more efficient in testing if there really is no difference in the two distributions (the Null Hypothesis). If
the probability of that magnitude is sufficiently small, then we would reject the Null Hypothesis, and
accept the Alternative Hypothesis instead. The bottom half of Figure 4 shows the decision criteria for
two-way variance comparisons.
For these last two tests, however, the assumption that the data comes from distributions that are normally
distributed is very important. In fact, we should assess how normally our data are distributed prior to
conducting either test.
Testing for a Normal Distribution
The idea behind the Normal Probability Plot is that if we structured a scatter plot of the sorted data values
and the probability of encountering those data values, that the plot would look roughly linear if the true
distribution were Normally distributed. Further, if two distributions are plotted on the same graph, since
the slope of each data set on a Normal Probability Plot is proportional to the variance, parallel plots
should have similar variances.
Normal probability plots may be done on special paper, may be done manually, or can easily be done
with Microsoft's Excel spreadsheet (an example of such a spreadsheet is available on disk). The
construction and evaluation steps are outlined below:
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1. Record raw data and count the observations (obtaining n)
2. Set up a column of values (from 1 to j), where j is an index
3. Compute (zj) for each j value:
(zj) = (j - 0.5)/n
4. Obtain a zj value for each (zj) from Standard Normal Table (find the entry ((zj) in the table, then
read index value (zj), or use the Excel Norminv() function)
5. Set up a column of observed data, sorted into increasing values
6. Construct a scatter plot of zj values versus sorted data values
7. Approximate the slope with a sketched line at the 25% and 75% data points
Evaluating a Normal Probability Plot
1. Assess the Equal-Variance and Normality assumptions:
Data from a Normal sample should tend to fall along the line, so if a “fat pencil” covers almost all of
the points, then a normality assumption is supported
The slope of the line reflects the variance of the sample, so equal slopes support the equal variance
assumption
2. Theoretically, a sketched line should intercept the zj = 0 axis at the mean value, so if the two plots
intersect at that point, they should have the same mean
3. Practically:
Close is good enough for comparing means
Closer is better for comparing variances
4. If the slopes differ much for two samples, use a test that assumes the variances are not the same
Summary of Hypothesis Testing Steps
State the null hypothesis (H0) that test statistic .
Choose an alternative hypothesis HA from one of the alternatives:  ,  , or .
Choose a significance level of the test (.
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 state the 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.
1.
2.
3.
4.
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Page 5 of 8
x – 0 – 0
Figure 4. Decision Chart for Hypothesis Testing.
Population Variance (2) is known
(use z-test)
Is the
Population
Mean known?
Yes (use 0)
z0 =
0
n
No (use x2)
x1 – x2 – 0
z0 =
12
22
+
Means
(z or t)
What is known
about the
Variation?
Population Variance
(2) is unknown
(use S2 & t-test)
Is the
Population
Mean known?
n1
Yes (use 0)
x1 – 0 – 0
t0 =
S
n
No
Are the
Variances
similar?
Kind of
comparison
?
n2
Yes
(use pooled S2)
x1 – x2 – 0
t0 =
1
Sp
1
+
n1
No
(use non-pooled S2)
n2
x1 – x2 – 0
t0 =
S12
S22
+
n1
Paired Comparison
(use difference in means & t-test)
d – 0
t0 =
Sd
n
Variances
(2
or F)
Is the Population
Variance known?
Yes
(use 0 & 2-test)
(n – 1) S2
 =
2
02
0
No
(use S2 & F-test)
S12
F0 =
S22
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n2
COMPARISON OF MEANS
Condition
Dist.
Estimator 1
Estimator 2
Variance
Known
(σ2)
Z
Mean
Known
(µo)
------------
Mean Unknown
(y)
Test Statistic
HA
Criteria
y - o
  o
Zo  Z 
  




 n
  o
  o
Z o  Z
y1  y 2
1   2
Zo  Z 
 12
1   2
1   2
Z o  Z 
Zo 
------------
Zo 
n1
Variance
Unknown
(S2)
t
Mean
Known
(µo)
Mean Unknown
(y)
------------
n2
y - o
to 
Similar Variances
 12   22

 22
 S 




 n
y1  y 2
to 
1
1

n1 n 2
Sp
Dissimilar Variances
 12   22
to 
Paired
Comparison
Difference In
Means
di  d1  d2 
i 1
n
i
to 
 Sd

 n

2
Z o  Z
to  t ,
n-1
to  t ,
t o  t ,
v
2
Where S p 
  2  2 
1

 2 
 n1
n2 


n 1
 S 


 n 



1
1
y1  y 2   t  , v  S p



n1 n 2
 2





(n1  1) S12  (n 2  1) S 22
and v  n1  n 2  2
n1  n 2  2
1   2
t o  t ,
1   2
t o  t ,
v
v



S2 S2 

y1  y2   t  , v  S p 1  2 


n1
n2 
 2 

2
 S12 S 22 



 n1 n2 

Where v  
2
2
 S12 
 S 22 




 n1 



   n2 
n1  1
n2  1
to  t ,

 S d 


d  t
 , n 1  n 
 2

n-1
2




Page 7 of 8

y  t ,

 2
  


 n 

n 1
n
Where S d 
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
y1  y 2   Z 

 2
2
1   2
d  0
d
Z o  Z
t o  t , n 1
S12 S 22

n1 n 2
n

y Z

 2
2
  o
  o
y1  y 2
Sample Mean Of
Differences
d
  o
Confidence Interval
 (d
i
 d )2
i 1
n 1
COMPARISON OF VARIANCES
Condition
Variance
Known
Dist.

( )
Test Statistic
 o2 
2
o
HA
2
(n  1) S
 o2
2
 2   o2
S
2
1
, S 22

F
S
Fo 
S
2
1
2
2
  
2
2
, n 1
or 
2
o
Confidence Interval

 2   o2
 o2  12 ,n1
 
    ,n1
2
Variance
Unknown
Criteria
2
o
 12   22
 
2
1
2
o
2
o
Fo  F
2
,n1 1,n2 1
 12   22

1 ,n1 1,n2 1
2
2
1, n 1 1
1
F , v2 , v1
NORMALITY ASSUMPTION IS CRITICAL
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(n  1) S 2 (n  1) S 2
, 2
2

2
, n 1


1 , n 1
2
or Fo  F
Fo  F ,n1 1,n2 1
NOTE: F1 , v1 , v2 

1 , n 1
2
2
Fo  F ,n
2
2
2
S12
S 22

 S12
F 

 1 ,n2 1,n1 1  , S 2
 2
 2


 F

 ,n2 1,n1 1 
 2
