chapter 7 statistical inference: estimating a population mean
TRANSCRIPT
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Chapter 7
Statistical Inference: Estimating a Population Mean
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Statistical Inference
Statistical inference is the process of reaching conclusions about characteristics of an entire population using data from a subset, or sample, of that population.
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Simple Random Sampling
Simple random sampling is a sampling method which ensures that every combination of n members of the population has an equal chance of being selected.
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Figure 7.1 A Table Of Uniformly
Distributed Random Digits 1 6 8 7 0 5 3 4 9 9 2 9 4 8
7 6 4 3 9 0 5 3 6 4 7 3 6 6
5 9 2 6 8 1 8 0 1 8 1 7 1 8
0 4 1 4 5 9 2 0 6 3 2 5 2 7
0 2 6 1 3 2 4 3 8 3 2 8 5 1
4 8 3 3 4 0 2 8 6 5 5 8 0 7
1 0 2 6 6 1 0 1 1 4 6 5 8 3
4 6 3 6 4 8 5 6 2 4 5 4 4 0
5 5 9 9 0 8 6 1 9 1 0 5 4 1
8 3 5 1 5 1 5 8 6 6 1 7 7 1
7 8 1 0 6 5 6 9 1 0 7 1 3 0
2 8 4 1 7 4 2 8 8 9 4 6 9 7
1 3 1 1 4 2 9 4 6 9 8 4 9 5
5 1 6 4 4 8 6 0 3 2 1 2 5 8
5 3 1 0 4 6 9 9 6 1 8 2 8 5
2 9 1 4 9 6 2 8 1 5 4 2 9 0
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Sample Data
Sample Population Hrs. of Study Member ID Time(x)
1 1687 20.0 2 4138 14.5 3 2511 15.8 4 4198 10.5 5 2006 16.3. . .. . .. . .49 1523 12.6
50 0578 14.0
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Figure 7.2 Bar Chart Showing the Population Study Time Distribution
x
P(x)
1/4 = .25
Study Time
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Figure 7.3 Sampling Distribution for for Our Small-Scale Illustration
P( )
15 .167
20 .167
25 .333
30 .167
35 .167
x x
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Figure 7.4 Bar Chart Showing the
Sampling Distribution of x
15 20 25 30 35 x
.167
.333
P( )x
Sample Mean Study Time (hrs)
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The Sampling Distribution of the Sample Mean
The sampling distribution of the sample mean is the probability distribution of all possible values of the sample mean, , when a sample of size n is taken from a given population.
x
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Key Sampling Distribution Properties
• For large enough sample sizes, the shape of the sampling distribution will be approximately normal.
• The sampling distribution is centered on , the mean of the population.
• The standard deviation of the sampling distribution can be computed as the population standard deviation divided by the square root of the sample size.
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Figure 7.5 The Shape of the Sampling Distribution When Sample Size is Large (n > 30)
x
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Central Limit Theorem
As sample size increases, the sampling distribution of the sample mean rapidly approaches the bell shape of a normal distribution, regardless of the shape of the parent population.
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Figure 7.6 Implications of the Central Limit Theorem
n = 2
n = 5
n = 30
x
x x x
x
xx
xx
x
x
The Sampling Distribution of the Sample Mean
x
Population Shapes
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Small Samples
In small sample cases (n<30), the sampling distribution of the sample mean will be normal if the shape of the parent population is normal.
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Figure 7.7 The Center of the Sampling
Distribution of the Sample Mean
E( ) = xx
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Standard Deviation of the (7.1) Sampling Distribution of the
Sample Mean
n
x
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Figure 7.8 Standard Deviation of the Sampling Distribution of the Sample Mean
n
x
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Figure 7.9 Sampling Distribution of the Sample Mean for Samples of Size n = 2, n = 8, and n = 20 Selected from the Same Population
Population Distribution
Sampling Distribution
n = 2
n = 20
n = 8
x
x
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Standard Deviation of the Sampling (7.2) Distribution of the Sample Mean (When sample size is a large fraction of the population size)
1
N
nN
n
x
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Interval Estimate of (7.3)
a Population Mean
n
xz
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Factors Influencing Interval Width
1. Confidence—that is, the likelihood that the interval will contain m. A higher confidence level will mean a larger z, which, in turn, will mean a wider interval.
2. Sample size, n. A larger sample size will produce a tighter interval.
3. Variation in the population, as measured by. The
greater the variation in the population values, the wider the interval.
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Figure 7.10 Intervals Built Around Various Sample Means from the Sampling Distribution
3x2x
4x5x
1x
x
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Figure 7.11 Standard Error vs.
Margin of Error
x + z x
Margin of Error
Standard Error
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Margin of Error
The margin of error in an interval estimate of measures the maximum difference we would expect between the sample mean and the population mean at a given level of confidence.
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Figure 7.12 General Comparison of the t and Normal Distributions
t distribution
Normal distribution
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Interval Estimate of (7.4) When s Replaces
x ts
n
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Figure 7.13 Comparison of the t and Normal Distributions as Degrees of Freedom Increase
Normal Distribution t with 15 degrees of freedom
t with 5 degrees of freedom
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Basic Sample Size Calculator (7.5)
2
E
zn
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Sample Size when (7.6) n/N > .05
N
nn
n
1