Introduction to the Normal Distribution (Dr. Monticino)

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Introduction to the Normal Distribution (Dr.
Monticino)
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Assignment SheetMath 1680

• Read Chapter 5
• Assignment 5
• Due Monday, Feb. 21st
• Chapter 5
• Set A 1,2
• Set B 1,2,3,5
• Set C 1,2,3
• Set D 1,2,3,4,5
• Set E 1,2,3
• Review exercises 1,2,3,4,7,8

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Overview

• Motivation
• Calculations

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The Normal Distribution

• Histograms for a variety of data often have a
  bell curve shape
• Because of this empirical observation and
  theoretical results, often approximate the
  distribution of a population by a normal
  distribution
• Recall area of bars in histogram give percentage
  in a group.
• Similarly, the area under a specified portion of
  the normal distribution curve gives the
  percentage in that range

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Normal Distribution Approximation
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The Normal Distribution

• The standard normal curve is symmetric about 0
• 50 of the area is to the left of 0 and 50 to
  the right of 0
• Analogous areas on opposite sides of 0 have the
  same area

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Standard Normal Distribution

• 68 of the area is within -1,1 95 between
  -2,2 and 99.7 between -3,3
• Other values are given by a normal table

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Converting to SD Units

• Examples
• Suppose that the average weight for U.S. men is
  175 lbs. and the standard deviation for weight is
  20 lbs.
• If a man weighs 190 lbs., how many standard
  deviation units away from the mean weight is he?
• A man is 3.2 standard deviations above the
  mean-weight. How heavy is he?
• Ninety-five percent of U.S. men weigh between
  what two values?

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Examples

• Find area under normal curve
• to the right of 1
• to the left of -.5
• to the left of .75
• between -1.5 and 1.5
• outside of -2 and 2

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Standard Normal Table
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Examples

• The birthweight for babies is approximately
  normally distributed with an average birthweight
  of 7.2 lbs. and a standard deviation of 1.1 lbs.
• A newborn baby weighing 9 lbs is how many
  standard deviation units away from the mean
  weight?
• What percent of newborns weigh between 8.3 and
  6.1 lbs.
• What percent of babies weigh more than 9 lbs?
• At what percentile of weight is a 9 lb. baby?
• What is the 25th percentile for newborns?
• Thirty percent of babies weigh between what two
  values (centered on the mean)?

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More Examples

• Nationally, SAT scores are distributed normally
  with an average score of 1000 and a standard
  deviation of 150
• What percent of students taking the SAT score
  above 1400
• What percent score between 900 and 1200
• What is the 95th percentile of SAT scores
• (Dr. Monticino)