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Unformatted text preview: na; Slide 29
University
29 Can We Assume Sampling from a
Normal Population?
If data are from a normal population,
there is a linear relationship between the
data and their corresponding Z values.
data Z= Y −µ σ Y =σ Z +µ If we plot y on the vertical axis and z on the horizontal
axis, the y intercept estimates µ and the slope estimates σ.
G. Baker, Department of Statistics
G.
University of South Carolina; Slide 30
University
30 How to Calculate Corresponding
How
ZValues
Order data
Order
Estimate percent of population below each
data point.
data
i − 0.5 Pi = n where i is a data point’s position in the ordered set
and n is the number of data points in the set. Look up ZValue that has Pi proportion of
Look
proportion
distribution below it.
distribution
G. Baker, Department of Statistics
G.
University of South Carolina; Slide 31
University
31 Normal Probability (QQ) Plot
Data set: Z Pi yi i 7 1 0.32 10 2 .375 4 2 .625 7 3 +1.15 4 .125 +0.32 2 1.15 .875 10 4 Normal QQ Plot
12
10
Data 8
6
4
2
0
1.5 1 0.5 0 0.5 1 1.5 Z values
G. Baker, Department of Statistics
G.
University of South Carolina; Slide 32
University
32 Normal Probability (QQ) Plot
QQ Plot with Data on Vertical Axis
16
14
12
10
8
6
4
2
0
3 2 1 0 1 2 This data is a random sample from a n(10,2) population.
G. Baker, Department of Statistics
G.
University of South Carolina; Slide 33
University
33 3 Normal Probability (QQ) Plot
QQ Plot with Data on Vertical Axis
16
14
12
10
8
6
4
2
0
3 2 1 0 1 2 3 G. Baker, Department of Statistics
G.
University of South Carolina; Slide 34
University
34 Estimation of the Mean G. Baker, Department of Statistics
G.
University of South Carolina Point Estimators
A point estimator iis a single number
s
point
calculated from sample data that is used to
estimate the value of a parameter.
estimate
Recall that statistics change value upon
repeated sampling of the same population while
parameters are fixed, but unknown.
parameters
Examples:
Examples: ˆ
p estimates p ˆ
µ = y estimates µ ˆ
ˆ = s estimates σ σ 2 = s 2 estimates σ 2
σ
G. Baker, Department of Statistics
G.
University of South Carolina; Slide 36
University
36 In General: θˆ is an estimator of the arbitrary parameter θ
What makes a “Good” estimator?
(1) Accuracy: An unbiased estimator of a
parameter is one whose expected value is equal
to the parameter of interest.
(2) Precision: An estimator is more precise if
its sampling distribution has a smaller
standard error*.
*Standard error is the standard deviation
G. Baker, Department of Statistics
G.
for the sampling distribution.
University of South Carolina; Slide 37
University
37 Unbiased Estimators
For normal populations, both the sample
mean and sample median are unbiased
estimators of µ.
Sampling Distributions for Mean and Median mean
median 8 6 4 2 0
µ 2 4 6 8 G. Baker, Department of Statistics
G.
University of South Carolina; Slide 38
University
38 Most Efficient Estimators
If you have multiple unbiased estimators, then you
choose the estimator whose sampling distribution
has the least variation. This is called the most
efficient estimator.
effi...
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 Fall '13
 Wang
 Statistics

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