Make a Diagram 2/27/08
Make a Diagram
Means
Interval for
Mean
Confidence
Interval
  x  t 2 s x
Hypotheses
Test Ratio
H0 :   0
t
H1 :    0
df  n  1 or
  x  z 2  x
z
x  0
or
sx
x  0
x
Critical Value
xcv   0  t  2 s x
sx 
H 0 : D  D0
D  d  t 2 s d
H 1 : D  D0 ,
(See table 3 for
df and s d )
D  1   2
D  d  z 2 s d
t
d  D0
or
sd
z
d  D0
sd
or
n
xcv   0  z 2  x
x 
Difference
between Two
Means
s

n
d cv  D0  t  2 s d
or
d cv  D0  z 2 sd
d  x1  x 2
If H 1 :    0 or H 1 : D  D0 ( D0 is usually zero), we have a two-sided test. If H 1 :    0 or
H 1 : D  D0 , we have a left-sided test. If H 1 :    0 or H 1 : D  D0 , we have a right-sided test.
Make a diagram. This diagram could look like a Normal curve, even when the distribution is not
symmetrical.
Confidence Interval: The idea is to draw the confidence interval and to reject the null hypothesis
for a test of one mean if  0 is not on the confidence interval
or for a test of two means if D0 (usually zero) is not on the confidence interval.
Whether the confidence interval is one or two-sided, it will include x for a test of one mean and
d  x1  x 2 for a test of two means. x or d  x1  x 2 will be in the center of your diagram.
This means that for a left-sided test the confidence interval will be made by replacing ‘=’ by ‘  ’ and ‘  ’
by ‘+.’ The interval will start somewhat to the right of the center of the diagram and include the far left
corner of the diagram.
This also means that for a right-sided test the confidence interval will be made by replacing ‘=’ by ‘  ’ and
‘  ’ by ‘–.’ The interval will start somewhat to the left of the center of the diagram and include the far
right corner of the diagram.
Test Ratio: A diagram will always have zero at its center. Zero will never be in the ‘reject zone. The test
ratio will be compared to t  , t , t  , z  ,  z  ,  z as appropriate. Let us call your calculated value
2
2
of the ratio ẑ or tˆ .
For a left-sided test the ‘reject’ zone will be to the left of zero and will go from t or  z  to the far left
corner of the diagram. Reject the null hypothesis if ẑ or tˆ is in the ‘reject’ zone.
For a right-sided test the ‘reject’ zone will be to the right of zero and will go from t  or z  to the far right
corner of the diagram. Reject the null hypothesis if ẑ or tˆ is in the ‘reject’ zone.
P-value: The p-value is defined as the probability of getting results as extreme or more extreme than those
actually observed. It is usually calculated by getting a test ratio. Let us call your calculated value of the
ratio ẑ or tˆ .
For a left-sided test, p  value  Pz  zˆ  or p  value  P t  tˆ .
For a right-sided test, p  value  Pz  zˆ  or p  value  P t  tˆ .
 
 
1
Make a Diagram 2/27/08
For a 2-sided test, you can take the smaller of p  value  2Pz  zˆ  and p  value  2Pz  zˆ  or the
smaller of p  value  2 P t  tˆ and p  value  2 P t  tˆ .




To do a conventional hypothesis test, reject the null hypothesis if the p-value is below the significance
level.
Critical value for the sample mean or the difference between two sample means: The critical value is
used to find a ‘reject’ zone. The null hypothesis mean,  0 , or the null hypothesis difference between two
means, D0 , is never in the reject zone and is always in the center of the diagram. The critical value is
compared to x or d  x1  x 2 .
This means that for a left-sided test the ‘reject’ zone will be to the left of  0 (the mean from the null
hypothesis) or D0 and will be found by replacing and ‘  ’ by ‘–.’ The ‘reject’ zone will start somewhat to
the left of the center of the diagram and include the far left corner of the diagram.
This also means that for a right-sided test the ‘reject’ zone will be to the right of  0 (the mean from the
null hypothesis) and will be found by replacing and ‘  ’ by ‘+.’ The ‘reject’ zone will start somewhat to the
right of the center of the diagram and include the far right corner of the diagram.
Proportions
Interval for
Proportion
Confidence
Interval
p  p  z s p
2
sp 
pq
n
Hypotheses
Test Ratio
H 0 : p  p0
z
H1 : p  p0
p  p0
p
q  1 p
Difference
between
proportions
q  1 p
p  p  z 2 s p
p  p1  p 2
s p 
p1 q1 p 2 q 2

n1
n2
H 0 : p  p 0
H 1 : p  p 0
z
p  p 0
 p
p 0  p 01  p 02
If p  0
or p 0  0
 p 
p 01q 01 p 02 q 02

n1
n2
s
Critical Value
pcv  p0  z 2  p
p0 q0
n
q0  1  p0
p 
pcv  p0  z 2  p
If p  0
 1
1 


n
n
2 
 1
 p  p 0 q 0 
p0 
n1 p1  n 2 p 2
n1  n 2
Or use p
If you replace  with p or D with p and always use z and never t , the discussion of means should
explain this pretty well.
2
Make a Diagram 2/27/08
Variances
Interval for
Confidence
Interval
VarianceSmall Sample
2 
VarianceLarge Sample
 
Ratio of Variances
1 , DF2
F1DF


2
1
FDF1 , DF2
2
Hypotheses
Test Ratio
H 0 :  2   02
2 
n  1s 2
.25 .5 2 
H1: :  2   02
s 2DF 
H 0 :  2   02
 z 2  2DF 
H1 : 
 22 s22 DF1 , DF2

F

 12 s12 .5  .5  2 
DF1  n1  1
2
Critical Value
n  1s 2
 02
z 
2  2  2DF   1
  02
H0 : 12   22
F DF1 , DF2 
H1 : 12   22
2
s cv

s cv 
 .25 .5 2  02
n 1
 2 DF
 z  2  2 DF
s12
s 22
and
DF2  n 2  1
F DF2 , DF1 
 2

.5  .5   2    or
1  
2

s 22
s12
Confidence Intervals: The confidence interval formulas for two-sided intervals are explained in
Confidence Limits and Hypothesis Tests for Variances and the outline documents extracted from them.
For actual hypothesis tests, the only method commonly used is a Test Ratio. For a small sample you are
using a diagram of the  2 statistic with n  1 degrees of freedom. The center of the diagram will be near
the mean of  2 which is equal to the number of degrees of freedom. This will never be in a ‘reject’ zone.
For a single variance, just as above, H1 :  2   02 or H 1 :    0 is a two sided test. The lower ‘reject’
zone will be below  12 2 . (For example, if n  25 and   .05 this will be below the .975 value in the
df  24 part of the chi-square table. ) The upper ‘reject’ zone will be above  22 . (For example, if
n  25 and   .05 this will be above the .025 value in the df  24 part of the chi-square table. ) If your
computed value of  2 
n  1s 2
 02
falls in one of the ‘reject’ zone, reject the null hypothesis.
For a single variance, just as above, H1 :  2   02 or H 1 :    0 is a left-sided test. The ‘reject’ zone will
be below 12 . (For example, if n  25 and   .05 this will be below the .95 value in the df  24 part of
the chi-square table.) If your computed value of  2 
n  1s 2
 02
falls in the ‘reject’ zone, reject the null
hypothesis.
For a single variance, just as above, H1 :  2   02 or H 1 :    0 is a right-sided test. The ‘reject’ zone
will be above  2 . (For example, if n  25 and   .05 this will be above the .05 value in the df  24 part
of the chi-square table.) If your computed value of  2 
hypothesis.
For a large sample, compute  2 
n  1s 2
 02
n  1s 2
 02
falls in the ‘reject’ zone, reject the null
and make it into a value of z  2  2  2df  1 . Then treat it
like a z test ratio for the mean or proportion.
3
Make a Diagram 2/27/08
For a ratio of variances, because the F table is set up for right-sided tests, all tests should be done as rights2
sided tests. If we have H1 :  12   22 or H 1 :  1   2 , we have a true right-sided test. Test the ratio 12
s2

n1 1,n2 1
against the critical value F
. Your diagram should show a center at 1 and a ‘reject’ zone above

Fn1 1,n2 1 (For example, if n1  25 , n 2  20 and   .05 this will be above the .05 value in the df1  24 ,
df 2  19 part of the F table.) Reject the null hypothesis if your value of
s12
s 22
falls in the ‘reject’ zone.
If we have H1 :  12   22 or H 1 :  1   2 , we have a left-sided test. Think of it as a right-sided test with
H1 :  22   12 or H 1 :  2   1 . Test the ratio
s 22
s12
against the critical value Fn2 1,n2 1 . Your diagram
should show a center at 1 and a ‘reject’ zone above Fn2 1,n1 1 (For example, if n1  25 , n 2  20 and
  .05 this will be above the .05 value in the df1  19 , df 2  24 part of the F table.) Reject the null
hypothesis if your value of
s 22
s12
falls in the ‘reject’ zone.
If you have to do a two-sided test, think of it as two one-sided tests. Test the ratio
24,19 ). Then test the ratio
value Fn1 1,n2 1 (for example F.025
2
s 22
s12
s12
s 22
against the critical
against the critical value Fn2 1,n2 1 for
2
19, 24 ). If either of these two tests results in a rejection, reject the null hypothesis. In practice,
example F.025
only one of these tests actually needs to be done, since if
s12
s 22
is above one,
s 22
s12
will be below 1 and critical
values from the F table cannot be below 1.
4