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Unformatted text preview: STAT503 — Spring 2009 Lecture Notes: Chapter 4 1 Chapter 4: The Normal Distribution September 9, 2009 4.1 Introduction • The normal curve is the canonical symmetric bellshaped density curve. • It is probably the most important distribution in probability and statistics. • The assumptions in using it as a model are not as restrictive as for binomial distribution. • It is for continuous data, not discrete data; hence we have a smooth density curve instead of a histogram. item Areas under the curve indicate probability. It describes a wide variety of data very well. Some examples are: • measurement errors, • test scores, • heights, yields, • body temperatures, • levels in blood samples (e.g. cholesterol), • etc. 4.2 The Normal Curves • The term “Normal” distribution actually refers to a family of distributions with similar shapes. • They are completely characterized by the (population) mean μ and the (popula tion) standard deviation σ . • We use the notation Y ∼ Normal( μ,σ ) or N ( μ,σ ) to indicate that Y has a normal distribution with mean μ and standard deviation σ ( note: many books use σ 2 as the parameter instead of σ ). Chapter4.tex; Last Modified: September 9, 2009 (W. Sharabati) STAT503 — Spring 2009 Lecture Notes: Chapter 4 2 • All normal densities look the same regardless of μ and σ , which only determine the scale. The 68%95%99.7% Rule Recall our general rule for bell shaped histograms: – Approximately 68% of the observations lie within 1 SD of the mean. – Approximately 95% of the observations lie within 2 SD of the mean. – Approximately 99.7% of the observations lie within 3 SD of the mean. These percentages actually come form the normal density... Chapter4.tex; Last Modified: September 9, 2009 (W. Sharabati) STAT503 — Spring 2009 Lecture Notes: Chapter 4 3 The 68%95%99.7% Rule 4.3 Areas Under a Normal Curve A particularly nice feature of normal random variables is that they are preserved under linear transformations. If Y has a Normal( μ,σ ) distribution and Y = aY + b then Y has a Normal( μ ,σ ) distribution where μ = aμ + b and σ = aσ. In particular, if we set Z = Y μ σ , then Z has a Normal(0 , 1) distribution. Such Z is called a standard normal random variable. Note that when Y lies, e.g., 2 . 5 σ to the right of μ then Z equals 2 . 5. The scale of Z has no units and it is called the standardized scale . It shows how many standard deviations from the mean the Y variable is. Chapter4.tex; Last Modified: September 9, 2009 (W. Sharabati) STAT503 — Spring 2009 Lecture Notes: Chapter 4 4 The Y scale and the Z scale compared......
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This note was uploaded on 04/23/2011 for the course STAT 503 taught by Professor Staff during the Spring '08 term at Purdue.
 Spring '08
 Staff
 Normal Distribution, Probability

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