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7.2

7.2. The Standard Normal Distribution. Standard Normal. The standard normal curve is the one with mean μ = 0 and standard deviation σ = 1 We have related the general normal random variable to the standard normal random variable through the Z -score

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7.2

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  1. 7.2 The Standard Normal Distribution

  2. Standard Normal • The standard normal curve is the one with mean μ = 0 and standard deviation σ = 1 • We have related the general normal random variable to the standard normal random variable through the Z-score • In this section, we discuss how to compute with the standard normal random variable

  3. Standard Normal • There are several ways to calculate the area under the standard normal curve • What does not work – some kind of a simple formula • We can use a table (such as Table IV on the inside back cover) • We can use technology (a calculator or software) • Using technology is preferred

  4. Area Calculations • Three different area calculations • Find the area to the left of • Find the area to the right of • Find the area between

  5. Table Method • "To the left of" – using a table • Calculate the area to the left of Z = 1.68 • Break up 1.68 as 1.6 + .08 • Find the row 1.6 • Find the column .08 • (Table is IV on back cover) • The probability is 0.9535

  6. Table Method • "To the right of" – using a table • The area to the left of Z = 1.68 is 0.9535 • The right of … that’s the remaining amount • The two add up to 1, so the right of is 1 – 0.9535 = 0.0465

  7. “Between” • Between Z = – 0.51 and Z = 1.87 • This is not a one step calculation

  8. We want We start out with, but it’s too much We correct by Between • Between Z = – 0.51 and Z = 1.87

  9. Table • The area between -0.51 and 1.87 • The area to the left of 1.87, or 0.9693 … minus • The area to the left of -0.51, or 0.3050 … which equals • The difference of 0.6643 • Thus the area under the standard normal curve between -0.51 and 1.87 is 0.6643

  10. We want We delete the extra on the left We delete the extra on the right A different “Between” • Between Z = – 0.51 and Z = 1.87

  11. Different “Between” • Again, we can use any of the three methods to compute the normal probabilities to get • The area between -0.51 and 1.87 • The area to the left of -0.51, or 0.3050 … plus • The area to the right of 1.87, or .0307 … which equals • The total area to get rid of which equals 0.3357 • Thus the area under the standard normal curve between -0.51 and 1.87 is 1 – 0.3357 = 0.6643

  12. Z-Score • We did the problem: Z-Score  Area • Now we will do the reverse of that Area  Z-Score • This is finding the Z-score (value) that corresponds to a specified area (percentile)

  13. Z-Score • “To the left of” – using a table • Find the Z-score for which the area to the left of it is 0.32 • Look in the middle of the table … find 0.32 • The nearest to 0.32 is 0.3192 … a Z-Score of -.47

  14. Z-Score • "To the right of" – using a table • Find the Z-score for which the area to the right of it is 0.4332 • Right of it is .4332 … left of it would be .5668 • A value of .17

  15. Middle Range • We will often want to find a middle range, to find the middle 90% or the middle 95% or the middle 99%, of the standard normal • The middle 90% would be

  16. Middle • 90% in the middle is 10% outside the middle, i.e. 5% off each end • These problems can be solved in either of two equivalent ways • We could find • The number for which 5% is to the left, or • The number for which 5% is to the right

  17. Middle • The two possible ways • The number for which 5% is to the left, or • The number for which 5% is to the right 5% is to the right 5% is to the left

  18. Common Z-Scores • The number zα is the Z-score such that the area to the right of zα is α • Some useful values are • z.10 = 1.28, the area between -1.28 and 1.28 is 0.80 • z.05 = 1.64, the area between -1.64 and 1.64 is 0.90 • z.025 = 1.96, the area between -1.96 and 1.96 is 0.95 • z.01 = 2.33, the area between -2.33 and 2.33 is 0.98 • z.005 = 2.58, the area between -2.58 and 2.58 is 0.99

  19. Terminology • The area under a normal curve can be interpreted as a probability • The standard normal curve can be interpreted as a probability density function • We will use Z to represent a standard normal random variable, so it has probabilities such as • P(a < Z < b) • P(Z < a) • P(Z > a)

  20. Calculator Method • "To the left of" – using a calculator • Calculate the area to the left of Z = 1.68 • Normalcdf(small number, z,0,1) • 2ndVars • Normalcdf( • The probability is 0.9535

  21. Calculator Method • "To the right of“ 1.68 – using a calculator • Normalcdf(Z, big number,0,1) • 2ndVars • Normalcdf( • 0.0465

  22. Between • Between Z = – 0.51 and Z = 1.87 • Normalcdf(low,high,0,1) • Normalcdf(-.51,1.87,0,1) .6642

  23. Z-Score • “To the left of” – using a Calculator • Find the Z-score for which the area to the left of it is 0.32 • InvNorm(.32,0,1) • Z-Score of -.47

  24. Z-Score • "To the right of" – using a calculator • Find the Z-score for which the area to the right of it is 0.4332 • Find the Complement of .4332 (1-.4332) • InvNormthat number • InvNorm(.5668,0,1) • A value of .1682

  25. Fun Stuff • Spend Time on this stuff…there is a lot to remember and keep organized! • Practice makes perfect!

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