Download sample - Milan C-2

Review of Statistical Terms
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Population
Sample
Parameter
Statistic
Population
the set of all measurements
(either existing or conceptual)
under consideration
Sample
a subset of measurements from a
population
Parameter
a numerical descriptive measure
of a population
Statistic
a numerical descriptive measure of
a sample
We use a statistic to make
inferences about a population
parameter.
Principal types of inferences
• Estimate the value of a population
parameter
• Formulate a decision about the value of a
population parameter
Sampling Distribution
a probability distribution for the
sample statistic we are using
Example of a Sampling
Distribution
Select samples with two elements
each (in sequence with
replacement) from the set
{1, 2, 3, 4, 5, 6}.
Constructing a Sampling
Distribution of the Mean for
Samples of Size n = 2
List all samples and compute the mean of each
sample.
sample:
mean:
sample:
mean
{1,1}
{1,2}
{1,3}
{1,4}
{1,5}
1.0
1.5
2.0
2.5
3.0
{1,6}
{2,1}
{2,2}
…
3.5
1.5
4
...
There are 36 different samples.
Sampling Distribution of the Mean
x
p
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
2/36
1/36
Sampling Distribution Histogram
6
36
1
36
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1
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1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
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Let x be a random variable
with a normal distribution with
mean  and standard
deviation
x . Let be the
sample mean corresponding
to random samples of size n
taken from the distribution .
Facts about sampling
distribution of the mean:
Facts about sampling
distribution of the mean:
• The x distribution is a normal
distribution.
Facts about sampling
distribution of the mean:
• The x distribution is a normal distribution.
• The mean of the x distribution is  (the
same mean as the original distribution).
Facts about sampling
distribution of the mean:
• The x distribution is a normal distribution.
• The mean of the x distribution is  (the
same mean as the original distribution).
• The standard deviation of the x
distribution is  n (the standard deviation
of the original distribution, divided by the
square root of the sample size).
We can use this theorem to
draw conclusions about
means of samples taken from
normal distributions.
If the original distribution is
normal, then the sampling
distribution will be normal.
The Mean of the Sampling
Distribution
x
The mean of the sampling
distribution is equal to the
mean of the original
distribution.
x  
The Standard Deviation of the
Sampling Distribution
x
The standard deviation of the
sampling distribution is equal to the
standard deviation of the original
distribution divided by the square
root of the sample size.

x 
n
The time it takes to drive
between cities A and B is
normally distributed with a mean
of 14 minutes and a standard
deviation of 2.2 minutes.
• Find the probability that a trip between
the cities takes more than 15 minutes.
• Find the probability that mean time of
nine trips between the cities is more
than 15 minutes.
Mean = 14 minutes, standard
deviation = 2.2 minutes
• Find the probability that a trip between
the cities takes more than 15 minutes.
15  14
z
 0.45
14 15
2.2
P( z  0.45)  1.00  0.6736  0.3264
Find this
area
Mean = 14 minutes, standard
deviation = 2.2 minutes
• Find the probability that mean time of
nine trips between the cities is more than
15 minutes.
 x    14
 2.2
x 

 0.73
n
9
Mean = 14 minutes, standard
deviation = 2.2 minutes
• Find the probability that mean time of nine
trips between the cities is more than 15
minutes.
Find this
area
15  14
z
 1.37
0.73
14 15
P( z  1.37 )  0.5  0.4147  0.0853