Chapter4print

Chapter4print - STAT503 — Spring 2009 Lecture Notes:...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 bell-shaped 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......
View Full Document

This note was uploaded on 04/23/2011 for the course STAT 503 taught by Professor Staff during the Spring '08 term at Purdue.

Page1 / 12

Chapter4print - STAT503 — Spring 2009 Lecture Notes:...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online