Stat 101 – Lecture 34
Inference for μ
• What is the mean heart rate
for all young adults?
• Use the sample mean heart
rate, y , to make inferences
about the population mean
heart rate, μ .
1
Sample Data
• Random sample of n = 25 young
adults.
• Heart rate – beats per minute
70, 74, 75, 78, 74, 64, 70, 78, 81, 73
82, 75, 71, 79, 73, 79, 85, 79, 71, 65
70, 69, 76, 77, 66
2
Summary of Data
• n = 25
• y = 74.16 beats
• s = 5.375 beats
• SE( y ) =
s
n
= 1.075 beats
3
Stat 101 – Lecture 34
Conditions
• Randomization condition:
random sample of 25.
• 10% condition: 25 is less than
10% of all young adults.
• Nearly normal condition: see
next slide.
4
Normal Quantile Plot
3
.99
2
.95
.90
1
.75
0
.50
.25
-1
.10
.05
-2
.01
-3
6
4
Count
8
2
60
65
70
75
80
85
90
Heart rate
5
Nearly Normal Condition
• Normal quantile plot – data
follows the diagonal line
representing a normal model.
• Box plot – symmetric.
• Histogram –symmetric and
mounded in the middle.
6
Stat 101 – Lecture 34
Confidence Interval for μ
y − tn*−1SE( y ) to y + tn*−1SE( y )
tn*−1 is from Table T
SE( y ) =
s
n
7
Table T
df
1
2
3
4
M
2.064
24
Confidence Levels 80%
90%
95%
98%
99%
8
Confidence Interval for μ
y − tn*−1SE( y ) to y + tn*−1SE( y )
74.16 ± 2.064(1.075)
74.16 − 2.22 to 74.16 + 2.22
71.94 beats to 76.38 beats
9
Stat 101 – Lecture 34
Interpretation
• We are 95% confident that the
population mean heart rate of
young adults is between
71.94 bpm and 76.38 bpm
10
Interpretation
• Plausible values for the population
mean.
• 95% of intervals produced using
random samples will contain the
population mean.
11
JMP:Analyze – Distribution
Mean
Std Dev
Std Err Mean
Upper 95% Mean
Lower 95% Mean
N
74.16
5.375
1.075
76.38
71.94
25
12
Stat 101 – Lecture 34
Test of Hypothesis for μ
• Could the population mean
heart rate of young adults be
70 beats per minute or is it
something higher?
13
Test of Hypothesis for μ
• Step 1: State your null and
alternative hypotheses.
H 0 : μ = 70
H A : μ > 70
14
Test of Hypothesis for μ
• Step 2: Check conditions.
–Randomization condition, met.
–10% condition, met.
–Nearly normal condition, met.
15
Stat 101 – Lecture 34
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
16
Summary of Data
• n = 25
• y = 74.16 beats
• s = 5.375 beats
• SE( y ) =
s
= 1.075 beats
n
17
Value of Test Statistic
y − μ0 74.16 − 70
=
SE( y )
1.075
t = 3.87
t=
Use Table T to find the P-value.
18
Stat 101 – Lecture 34
Table T
One tail probability 0.10
0.05
0.025
0.01
0.005 P-value
df
1
2
3
4
M
24
2.064
2.492
2.797 3.87
The P-value is less than 0.005.
19
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.
20
Test of Hypothesis for μ
• Step 5: State your conclusion
within the context of the
problem.
• The mean heart rate of all
young adults is more than 70
beats per minute.
21