Download Goal: Use sample data to estimate a population

Goal: Use sample data to estimate a population characteristic
1. point estimate
2. confidence interval estimate
Limitation of point estimate
A point estimate by itself does not convey any information about the
accuracy of the estimate.
Summary for CI
General Format: estimate ± margin of error
Note: Margin of error is also called bound on the error of estimation.
CI: an interval that is computed from sample data and provides a
range of plausible values for a population characteristic
Confidence level: success rate of the method used to construct the
confidence interval.
Most common confidence level is 95%.
Interpretation of CI
Example: a 90% CI for π, the proportion of students in York
University who own a computer, is (0.36, 0.58).
• Interpretation of intervalà we can be 90% confident that the
true proportion of students in York U who own a computer, is
between 36% and 58%.
• Interpretation of confidence levelà We have used a method to
produce this estimate that is successful in capturing π 90% of
the time. (The confidence level 90% refers to the method used
to construct the interval rather than to any particular interval)
1
Summary of t distribution
Let X1, _X2, …, Xn constitute a random sample from a normal
population distribution. Then the probability distribution of the
x−µ
standardized variable t = s / n is the t distribution with (n-1) df.
• The Student's t distribution is very similar to the standard
normal distribution.
• It is bell shaped and symmetric about its 0.
• Each t curve is more spread out than the z curve.
• As the number of df increases, the spread of the
corresponding t-curve decreases.
• As the number of df increases, the t distribution approaches
the normal distribution.
Comparison of normal and t distibutions
df = 2
df = 5
df = 10
df = 25
Normal
-4
-3
-2
-1
0
1
2
3
4
2
Example 1
Based on a random sample of 100 workers, it is reported that a
mean annual radiation exposure of 0.481. If the true population
standard deviation is 0.35, what is the 95% CI for the true
population mean radiation exposure?
Example 2
One article reported that the sample mean and standard deviation
for high school GPA for students enrolled at a large research
university were 3.73 and 0.45, respectively. Suppose that the mean
and standard deviation were based on a random sample of 900
students at the university. Construct a 95% CI for the mean high
school GPA for students at this university.
Example 3
Fat contents (in percentage) for 10 randomly selected hot dogs
were given as follows
25.2, 21.3, 22.8, 17.0, 29.8, 21.0, 25.5, 16.0, 20.9, 19.5
Assume that the fat content is approximately normal.
Use the data to construct a 90% CI for the true mean fat
percentage of hot dogs.
3
Central area captured:
Confidence level:
1
2
3
4
5
6
D
7
e
8
g
9
10
r
11
e
12
e
13
s
14
15
16
o
17
f
18
19
20
f
21
r
22
e
23
24
e
25
d
26
o
27
m
28
29
30
40
60
120
z critical values
0.80
0.90
0.95
0.98
0.99
0.998
0.999
80%
90%
95%
98%
99%
99.8%
99.9%
3.08
1.89
1.64
1.53
1.48
1.44
1.41
1.40
1.38
1.37
1.36
1.36
1.35
1.35
1.34
1.34
1.33
1.33
1.33
1.33
1.32
1.32
1.32
1.32
1.32
1.31
1.31
1.31
1.31
1.31
1.30
1.30
1.29
1.28
6.31
2.92
2.35
2.13
2.02
1.94
1.89
1.86
1.83
1.81
1.80
1.78
1.77
1.76
1.75
1.75
1.74
1.73
1.73
1.72
1.72
1.72
1.71
1.71
1.71
1.71
1.70
1.70
1.70
1.70
1.68
1.67
1.66
1.645
12.71
4.30
3.18
2.78
2.57
2.45
2.36
2.31
2.26
2.23
2.20
2.18
2.16
2.14
2.13
2.12
2.11
2.10
2.09
2.09
2.08
2.07
2.07
2.06
2.06
2.06
2.05
2.05
2.05
2.04
2.02
2.00
1.98
1.96
31.82
6.96
4.54
3.75
3.36
3.14
3.00
2.90
2.82
2.76
2.72
2.68
2.65
2.62
2.60
2.58
2.57
2.55
2.54
2.53
2.52
2.51
2.50
2.49
2.49
2.48
2.47
2.47
2.46
2.46
2.42
2.39
2.36
2.33
63.66
9.92
5.84
4.60
4.03
3.71
3.50
3.36
3.25
3.17
3.11
3.05
3.01
2.98
2.95
2.92
2.90
2.88
2.86
2.85
2.83
2.82
2.81
2.80
2.79
2.78
2.77
2.76
2.76
2.75
2.70
2.66
2.62
2.58
318.29
22.33
10.21
7.17
5.89
5.21
4.79
4.50
4.30
4.14
4.02
3.93
3.85
3.79
3.73
3.69
3.65
3.61
3.58
3.55
3.53
3.50
3.48
3.47
3.45
3.43
3.42
3.41
3.40
3.39
3.31
3.23
3.16
3.09
636.58
31.60
12.92
8.61
6.87
5.96
5.41
5.04
4.78
4.59
4.44
4.32
4.22
4.14
4.07
4.01
3.97
3.92
3.88
3.85
3.82
3.79
3.77
3.75
3.73
3.71
3.69
3.67
3.66
3.65
3.55
3.46
3.37
3.29
4