Tests for a Single Mean

1

• Lesson 20
• Tests for a Single Mean

2
Test of H0 m m0, when s2 is known
(one-sample Z-test for the mean)
3
Test of H0 m m0, when s2 is known
(one-sample Z-test for the mean)
under H0
test statistic
4
H1 m gt m0
0
Reject if Z0 is relatively large
Rejection region
Z1-a
Reject H0 if Z0 gt Z1-a
P(Z0 gt critical value H0 true)
P(reject H0 H0 true)
a
5
H1 m gt m0
0
Rejection region
Z1-a
P(Z0 lt critical value H1 true)
P(fail to reject H0 H1 true)
b
6
H1 m gt m0
0
Rejection region
Z1-a
Power P(reject H0 H1 true)
1 - b
7
H1 m gt m0
0
p-value is the area under N(0,1) curve above
observed Z0
Rejection region
Z1-a
? reject H0
? p-value P(Z gt Z0)
Reject H0 if p-value lt a
8
H0 m m0 vs. H1 m gt m0
Reject H0 if Z0 gt Z1-a
p-value P(Z gt Z0)
H0 m m0 vs. H1 m lt m0
Reject H0 if Z0 lt Za -Z1-a
p-value P(Z lt Z0)
9
H0 m m0 vs. H1 m ? m0
Z0 gt Z1-a/2
Reject H0 if
or
Z0 lt Za/2 -Z1-a/2
Reject H0 if Z0 gt Z1-a/2
p-value 2?P( Z gt Z0 )
10
Hypothesis Testing Steps
1) Determine hypotheses
2) Decide on a
( .01 , .05 , .10 )
3 4) State rejection region, calculate test
statistic
(or)
Calculate test statistic and p-value
5) Make decision (reject or not reject)
6) Write conclusions (interpret results), in
the context of the problem
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Test of H0 m m0, when s2 is unknown
(one-sample t-test for the mean)
If the variance is known, the test statistic was
If the variance is unknown, the test statistic is
12
H0 m m0 vs. H1 m gt m0
Reject H0 if t0 gt t(n-1),1-a
p-value P(t(n-1) gt t0)
H0 m m0 vs. H1 m lt m0
Reject H0 if t0 lt -t(n-1),1-a
p-value P(t(n-1) lt t0)
H0 m m0 vs. H1 m ? m0
Reject H0 if t0 gt t(n-1),1- a/2
p-value 2?P(t(n-1) gt t0 )
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H0 m 7250
H1 m lt 7250
a .05
Reject H0 if t0 lt -t(14),.95
-1.761
4767
- 7250
-3.00
3204
Reject H0
Conclude mean white blood cell count among Humans
infected with E. canis is less than 7250, the
mean of uninfected humans.
14
H0 m 100
H1 m ? 100
a .05
Reject H0 if t0 gt t(17),.975
2.110
112.778
- 100
3.76
14.424
Reject H0
Conclude mean percent ideal body weight among
insulin-dependent diabetics is greater than 100.
15
SAS yields p-values for two-sided tests when
performing t-tests.
If you are doing a one-sided test, you have to
adjust the SAS results to get the actual p-value.
If the results are in the direction of H1
If the results are in the opposite direction of H1