Stat 101 – Lecture 35
Alternatives
H 0 : μ = μ0
H A : μ < μ0 , One tail prob (Pr < t )
H A : μ > μ0 , One tail prob (Pr > t )
H A : μ ≠ μ0 , Two tail prob (Pr > t )
1
JMP:Analyze – Distribution
• Test Mean
t-test
Hypothesized value
Actual Estimate
df
Std Dev
70 Test statistic
3.87
74.16 Prob > |t|
0.0007
24 Prob > t
0.0004
5.375 Prob < t
0.9996
2
Another Example
• Is the population mean octane
rating 90 or is it something
different?
3
Stat 101 – Lecture 35
Test of Hypothesis for μ
• Step 1: State your null and
alternative hypotheses.
H 0 : μ = 90
H A : μ ≠ 90
4
Test of Hypothesis for μ
• Step 2: Check conditions.
–Randomization condition, met.
–10% condition, met.
–Nearly normal condition, met.
5
Normal Quantile Plot
3
.99
2
.95
.90
1
.75
0
.50
.25
-1
.10
.05
-2
.01
-3
6
4
Count
8
2
87
88
89
90
91
92
93
Octane Rating
94
95
6
Stat 101 – Lecture 35
Test of Hypothesis for μ
• Step 3: Calculate the test statistic
and convert to a P-value.
t=
y − μ0
SE( y )
SE( y ) =
s
n
7
Summary of Data
• n = 40
• y = 90.9475
• s = 1.530
• SE( y ) =
s
= 0.2419
n
8
Value of Test Statistic
y − μ0 90.9475 − 90
=
SE( y )
0.2419
t = 3.92
t=
Use Table T to find the P-value.
9
Stat 101 – Lecture 35
Table T
Two tail probability 0.20
0.10
0.05
0.02
0.01
2.021
2.423
2.704
P-value
df
1
2
3
4
M
39
40
3.92
The P-value is less than 0.01.
10
Test of Hypothesis for μ
• Step 4: Use the P-value to reach
a decision.
• The P-value is very small,
therefore we should reject the
null hypothesis.
11
Test of Hypothesis for μ
• Step 5: State your conclusion
within the context of the
problem.
• The population mean octane
rating is not 90 but something
different.
12
Stat 101 – Lecture 35
Confidence Interval for μ
y − tn*−1SE( y ) to y + tn*−1SE( y )
90.9475 ± 2.021(0.2419 )
90.95 − 0.49 to 90.95 + 0.49
90.46 to 91.44
13
Interpretation
• We are 95% confident that the
population mean octane rating is
between
90.46 and 91.44
14
Interpretation
• This confidence interval agrees
with the test of hypothesis.
• 90 is not in the interval and so
must be rejected as a value for
the population mean.
15