Download STATISTICAL INFERENCE Statistical Inference Statistical Inference

STATISTICAL INFERENCE
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Sampling
Sampling distributions
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QMS 204: STATISTICS FOR MANAGERS
l Instructor: Moez Hababou
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Statistical Inference
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The use of random samples from a
population to make inferences about a
population
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Statistical Inference
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Two types of inference procedures
Estimation
Hypothesis Testing
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Random Sample
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All individuals or objects in the population
have a chance of being selected
The selection of any one individual or object
does not affect the selection probability for
any other individual or object
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Simple Random Sample
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A random sample selected in such a way that
all possible samples of the same size have the
same chance of being selected
Implies that all individuals have the same
chance of being selected for the sample
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Finite Populations and Sampling With
Replacement
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We assume that samples are drawn without
replacement and that the population is large
enough that the number of possible samples
of size n is still large
If the sampling fraction n/N ( where N is the
population size) is less than 5% then we
ignore the finite population effect
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Finite Populations and Sampling With
Replacement
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If the population is small and finite
corrections have to be made(not to be
discussed in this course)
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Population Distribution
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This is the distribution of all items in the
population of interest
It is characterized by the values of certain
parameters which are usually unknown
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Examples
Population Distribution
X the random variable
µ population mean
σ2 population variance
draw samples
Sample 1
of size n
µ
σ
X
Sample 2
of size n
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Sample Distribution
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This is the distribution of the items that
appeared in the sample
This information is used to derive the values
of sample statistics which are used to
estimate the unknown parameters
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Examples
Population Distribution
X the random variable
µ population mean
σ2 population variance
Sample
of size n
x1, x2, x3
…., x n ,
µ
σ
X
Sample distribution
x
x
s
S2
X
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Sampling Distribution
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This is the distribution of a sample statistic
over all possible samples of the same size
taken from the same population
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Examples
Sampling distribution of
x
µ
Sampling distribution of S 2
x
σ2
S2
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Statistical Inference For The Population
Mean
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Requires the sampling distribution of the
mean
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Sampling Distribution of The Sample
Mean
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This is the distribution of the sample mean
over all possible simple random samples of
the same size taken from the same population
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Normal Population Variance Known
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The sampling distribution of the mean is a
Normal distribution with mean µ and
standard deviation σ/√n
The standard deviation of the mean σ/√n is
called the standard error of the mean
As the sample size n increases the variation
of the sample means around the true mean
decreases
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Examples
Distribution of x
Small Sample
• more variation in
Large Sample
• less variation in
x
µ
x
σ/ n
x
µ
x
σ/ n
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Examples
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It has been determined that heating expenses
are normally distributed with a mean of
$500.00 and a standard deviation of $125.00.
a) What is the probability that in a random
sample of 25 accounts the average balance
could exceed $550?
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Examples
Z = 550 - 500 = 2
125/ 25
500
550
P[ Z > 2 ] = 0.0228
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Examples
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b) What is the probability that in a random
sample of 25 accounts the average would lie
between $475.00 and $540.00?
Z = 540 - 500 = 1.60
125/ 25
Z = 475 - 500 = -1.00
475
500
540
125/ 25
P[ -1 ≤ Z ≤ 1.60 ] = 0.7865
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Nonnormal Population
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Central Limit Theorem
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Central Limit Theorem
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In large samples the central limit theorem
states that the sample mean is normally
distributed with mean µ and standard
deviation σ/√n
Thus in large samples we need not be
concerned about whether the underlying
population is normal
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Examples
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A random sample of 64 residents of a
particular section of the city were asked to
indicate the number of hours per week they
watch television. The population standard
deviation is 16.
a) What is the probability that the sample
mean will exceed the population mean by 2
in absolute value?
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Examples
µ- 2 µ µ+2
Z=
(µ+2) -
µ
= 1
Z=
16 / 64
(µ-2) - µ
= -1
16 / 64
P[ |Z| > 1 ] = 2(0.1587)
= 0.3174
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Examples
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b) If the sample size were 144 what would be
the probability in a)?
µ- 2
µ
µ+2
Z=
(µ+2) -
Z=
(µ-2) -
µ
16/ 144
µ
16/ 144
= 1.5
= -1.5
P[ |Z| > 1.5 ] = 2(0.0668)
= 0.1336
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Sampling Distribution for the Mean
When Variance Unknown
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Population variance unknown in large
samples
Population variance unknown in small
samples
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Population Variance Unknown in Large
Samples
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By the central limit theorem in large samples
the statistic below has a standard normal
distribution
 x − µ


 s/ n 
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This distribution is exact if the population
being sampled is normal
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Population Variance Unknown in Small
Samples
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In small samples if the population variance is
unknown and the population is normal the
statistic below has a t distribution with (n-1)
degrees of freedom
 x − µ


 s/ n 
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If the population is not normal we do not
know the distribution of this statistic
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t-Distribution
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the t distribution is bell shaped and
symmetrical and looks very much like a
standard normal distribution
Normal distribution
t distribution
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t-Distribution
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the mean of the distribution is zero and the
variance is larger than the standard normal
variance of 1
as the number of degrees of freedom
increase, the variance of the t distribution
approaches 1
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t-Distribution
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the critical values of the t distribution are
larger than the corresponding critical values
for the standard normal but approach the
standard normal values as the number of
degrees of freedom rise
by the time d.f. = (n-1) = 30 the t distribution
and standard normal are very close
examine the critical values of t for α = 0.025
as the d.f. change
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Sampling For Proportions
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In the special case of a binomial population
we are interested in making inferences about
the binomial parameter π
This is a special case of the central limit
theorem
In large samples the sampling distribution of
the sample proportion p has a mean of π and
a standard deviation of √π (1-π)/n
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Sampling For Proportions
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The sample proportion p is equivalent to a
sample mean for a variable whose underlying
values are either 0 or 1
In the case of a sample proportion the sample
size needs to be large if the true proportion π
is close to zero or 1
A useful rule of thumb is that both n π and
n(1-π) should exceed 5
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Examples
If π = 0.10
n π > 5 ⇒ n > 50
If
π = 0.90
n(1- π) > 5 ⇒ n > 50
If π = 0.01
n π > 5 ⇒ n > 500
If
π = 0.99
n(1- π) > 5 ⇒ n > 500
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Examples
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Suppose that 60% of the population of
Alberta voters wish to eliminate video lottery
terminals. Suppose a sample of 100 voters
was selected and asked for their opinion.
What is the probability that less than 50% of
the sample will be in favor of eliminating the
terminals?
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Examples
Z = 0.50 - 0.60
= -2.04
(0 .60)(0 .40)
100
0.5
0.6
P[ Z < -2.04 ] = 0.0207
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