QMS Chapter8Examples (Fall2012).doc

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Ryerson University
Quantitative Methods
QMS 102
Jason Chin- Tiong Chan

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. Fall2012 Page#1 Business Statistics I –QMS102 Chapter8 Sampling Distribution of the Mean 1. The distribution of all possible sample means. N ∑ Xi Population mean μ = i=1 N The average of all sample means, μ X Standard Error of the Mean The value of the standard deviation of all possible sample means σ = σ X n The Central Limit Theorem For a large sample size, the sampling distribution of is approximately normal, irrespective of the shape of the population distribution. The mean and standard deviation of the sampling distribution ofare μ = μ and σ = σ X X n The sample size is usually considered to be large if ≥ 30 Finding the Z for the Sampling Distribution of the Mean X − μ X X − μ Z = σ = σ X n 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. Fall2012 Page#2 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
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