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Final

QMS 102 Study Guide - Final Guide: Central Limit Theorem, Sampling Distribution, Standard Deviation


Department
Quantitative Methods
Course Code
QMS 102
Professor
Jason Chin- Tiong Chan
Study Guide
Final

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Business Statistics I –QMS102 Chapter8
Chapter8 Sampling distributions
Outcomes
1. Discuss the importance of sampling and the main reasons for sampling
2. Explain the definition of sampling distributions
3. Explain the central limit theorem
4. Calculate the standard error of the mean
5. Use the central limit theorem to determine probability of selecting possible
sample means from a specified population
Sampling
1. Select a portion of the population that is most representative of the population
2. Provide sufficient information so that conclusions (inferences) can be drawn
about the characteristics of the population
Reasons for Sampling
1. To contact the whole population would often be time consuming
2. The cost of studying all the items in a population may be prohibitive
3. Some tests could be destructive in nature
If the wine tasters in Niagara-on-the –Lake drank all the wine to evaluate
the vintage, they would consume the entire crop, and none would be
available for sale
4. The sample results are adequate
Example: The Federal Government uses a sample of grocery stores
scattered throughout Canada to determine the monthly index of food
prices. The prices of bread, milk and other major food items are included
in the index. It is unlikely that the inclusion of all grocery stores in Canada
would significantly affect the index, since the prices of milk, bread, and
other major food usually do not vary by more than a few cents from one
chain store to another.
Sampling distributions
1. For any population data set, there is only one value of the population
mean and population standard deviation.
2. Different samples drawn from the same population may result in different
sample mean and sample standard deviation.
3. Sampling distribution is the distribution of all sample statistics.
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Business Statistics I –QMS102 Chapter8
Sampling Distribution of the Mean
1. The distribution of all possible sample means.
Population mean
N
X
N
i
i
=
=1
µ
The average of all sample means,
X
µ
Standard Error of the Mean
The value of the standard deviation of all possible sample means
n
X
σ
σ
=
The Central Limit Theorem
For a large sample size, the sampling distribution of
x
is approximately normal,
irrespective of the shape of the population distribution. The mean and standard
deviation of the sampling distribution of
x
are
µµ
=
X
and
n
X
σ
σ
=
The sample size is usually considered to be large if
30n
Finding the Z for the Sampling Distribution of the Mean
n
X
X
Z
X
X
σ
µ
σ
µ
=
=
Note:
The sampling distribution of the sample mean will follow a normal probability
distribution under two conditions:
1. When the samples are taken from population known to follow a normal
distribution. In this case, the size of the sample is not a factor.
2. When the shape of the population distribution is not known or the shape is
known to be non-normal but the sample contains at least 30 observations.
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Business Statistics I –QMS102 Chapter8
Example1
The mean rent for a one-bedroom apartment in downtown Toronto is $1200 per
month, with a standard deviation of $250. The distribution of the monthly costs
does not follow the normal distribution. In fact, it is positively skewed. What is the
probability of selecting a sample of 50 one-bedroom apartments and finding the
mean to be at least $950 per month?
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