The Normal Curve

Discussion

• Look at the shape of a histogram
• Uniform Distribution
• Poisson
• Normal or Gaussian
• In a normal distribution the mean, median and mode are all the same number.
• we will use μ for mean and σ for standard deviation
• This is the "bell shaped curve"
  • 50% of the data is below μ
  • 68% of the data is between μ-σ and μ+σ
  • 95% of the data is between μ-2σ and μ+2σ
• The z-score is a measure of how far a given data value is from the mean
  • z = (x-μ)/σ
  • A distribution has μ=50 and σ=5, find the z score for
    • 45 (45-50)/5 = -5/5 = -1
    • 47 (47-50)/5 = -3/5
    • 50 (50-50)/5 = 0
    • 51 (51-50)/5 = 1/5
    • 55 (55-50)/5 = 1
  • We use the z score to find the percentage of data between two values.
  • We use table 13,7 on page 800 to do this.
  • A z score of 1/5 (.2) has a value of A corresponding to .079
  • This means that the area under the curve from μ to 1/5 is 0.79
  • Or that 7.9% of the population is between the mean and 51.
  • To do this
    • Draw a picture, put in your scores
    • Calculate the z score(s)
    • Look up the scores in the table
    • Find the area of interest.
  • #32

    	z = 1.19,  A = .383
    	.5 for below  the mean plus .383 from mean to position
    	.883 or 83%
    	

  • #34, page 805

    	    z= 2.08 A=.481
    	    To the right of this we need to subtract from .5
    	    .5-.481 =.019
    	

  • 40,

    between z = -0.15    (A=.060)
    and     z = -0.82    (A=.294)
    
           Area = .294-.06 = .234
    	

  • 56

    μ = 1600
    σ = 100
    More than 1750 hours.
    
          z = x-μ/σ
    
          z = 1750-1600/100 = 150/100 = 1.5
    
          A = .433
          So .5-.433 = .067 or 6.7%
    	

Homework