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381
Inferences About the Mean-I
(Large Samples)
QSCI 381 – Lecture 21
(Larson and Farber, Sect 6.1)
Statistical Inference
381


Sample statistics can be used to
the value(s) of unknown
population parameter(s). We are making
inference about the unknown population
parameter(s) based on data.
We start by making estimates of the
population parameter :


when the sample size is large (30)
when the sample size is small.
Point Estimation
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


A
is a single value
for a population parameter.
The most unbiased point estimate of
the population mean  is the sample
mean x .
Warning: x is an estimate and not the
population mean – the difference
between the two is the uncertainty of
the estimate.
Example
381
An acoustic survey involved three passes of an area.
The average density during the first pass was 52.3.
12
Results for passes 2 and 3
10
x  50.4
x  51.0
Frequency
8
6
4
2
0
30
35
40
45
50
55
Density
60
65
70
75
80
Interval Estimation
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
An
is an interval, or
range of values, used to estimate a
population parameter.
41.0
44.6
48.2
51.8
55.4
Point estimate=52.3
Interval = (48.5, 58.1)
59.0
Level of Confidence
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
The
is the
probability that the interval estimate
contains the population parameter.
For n30, the sampling distribution of sample
means is a normal distribution. The level of
confidence, c, is the area under the standard
normal distribution between –zc and zc.
Hint: for zc=1.645, there is a 90% probability
that the interval estimate contains the
population mean. Why do I say this?
c
-zc
z=0
½(1-c)
zc
Extent of Error
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
Given a level of confidence, c, the
(margin of error or error tolerance)
is the greatest possible difference
between the point estimate and the
value of the parameter being
estimated:
E  zc  x  zc

n
Example-I
381

1.
2.
3.
4.
Use the data for the first pass of the acoustic survey
and a 95% level of confidence (i.e. zc=1.96) to find
the maximum error of estimate for the mean density.
The steps to calculate the maximum error of estimate
are:
Find the sample statistics n and x .
Specify  if known. Otherwise, if n30, find the
sample standard deviation, s, and use this as an
estimate of .
Find the level of z that corresponds to the confidence
level.
Find the maximum error of estimate of E.
Example-II
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
Application of these steps gives:
1.
n=50; x  52.28
2.
 is not known so we estimate it from the
sample:
1 n
2
s
x  x

n 1
i 1
i
 10.68
The critical value of z is 1.96.
2.
The maximum error of estimate E is:

10.68
E  zc
 1.96
 2.95
n
50
We are 95% confident that the maximum error of
estimate for the population mean is about
2.95 units of density.
1.
Confidence Intervals for the
Population Mean
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
A
population mean  is
for the
x E   x E


The probability that the confidence
interval contains  is c.
The 95% confidence interval for the
mean density is (52.28-2.95,
52.38+2.95) = (49.3, 55.2)
Confidence Intervals for the
Population Mean
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To compute a (95%) confidence
interval using EXCEL:

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
=AVERAGE(D2:D51)+STDEV(D2:D51)/SQRT(COUNT(
D2:D51))*NORMINV(0.025,0,1)
=AVERAGE(D2:D51)+STDEV(D2:D51)/SQRT(COUNT(
D2:D51))*NORMINV(0.975,0,1)
Note: The calculation is based on
STDEV and not STDEVP.
Example-III
( known to be 10)
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Application of the steps gives:
1.
n=50;  =10; x  52.28
2.
The critical value of z is 1.96.
3.
The maximum error of estimate E is:

10
E  zc
 1.96
 2.77
n
50
We are 95% confident that the maximum error of
estimate for the population mean is about
2.77 units of density.

Interpretation
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40
45
50
55
60
100 simulated confidence intervals for the
mean of acoustic survey density
Summary
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Given c find zc
Compute
x   xi
Is  known?
Yes
E  zc
n
i 1
1 n
2
s
 xi  x 

n  1 i 1

n
x E   x E
No
E  zc
s
n