The Normal Curve

• Called the normal or Gaussian distribution.
• The shape is the bell curve.
• Many data sets follow this distribution
  • Height of the population
  • Weight of the population
  • Time to Fail in break shoes
  • Life span of a refrigerator
  • lengths of fully grown boa constrictors.
• We will see many more in the examples.
• The normal curve describes a population, so we will us μ for mean and σ for standard deviation.
• The mean, median and mode of the normal curve are equal.
• This means that 50% of the population is below μ and 50% of the population is above μ
• σ determines how spread out the curve is.
• The Empirical Rule
  • 68% of the population is between μ-σ and μ+σ
  • Given a population with μ=100 and σ=12.
    • What percentage of the population has a score?
    • What percentage of the population has a score more than 100?
    • What percentage of the population has a score less than 100?
    • What percentage of the population has a score between 88 and 112?
    • What percentage of the population has a score less than 88 or more than 112?
    • What percentage of the population has a score between 88 and 100?
    • What percentage of the population has a score between 100 and 112?
    • What percentage of the population has a score less than 88
    • What percentage of the population has a score more than 112?
  • 95% of the population have a score between μ-2σ and μ+2σ
  • Do the above exercise with 76, 88, 100, 112, and 124
  • 99.7% of the population have a score between μ-3σ and μ+3σ
  • Do the above exercise with 64, 76, 88, 100, 112, 124 and 136
• z-scores
  • We will will need a table for scores that are not integer multiples of σ
  • To avoid having to produce infinitely many tables, we reduce all normal curves to one with μ=0 and σ=1
  • To do this, for any data in a normal distribution, we calculate the z-score.
  • z = (x-μ)/σ
  • For the above problem, what is the z-score for 100, 88, 112, 90, 106?
• Using z-scores to find percentage of a population
  • The tables on page 888 and 889 show the area, and thus percentage under the normal curve to the left of a given z-score.
  • look at z=0, z=-1, z=1, z=-2, z=2
• How can I find the percentage of the population within μ+/- 1/2σ?
• How can I find the percentage of the population within μ+/- 3/2σ?
• Do problems 27-81 page 895.