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SECTION 6.2

THE STANDARD NORMAL DISTRIBUTION. SECTION 6.2. NORMAL PROBABILITY DISTRIBUTION. The mean, median and mode are the same. The distribution is bell-shaped and symmetrical around the mean. The total area under the curve is equal to 1.

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SECTION 6.2

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  1. THE STANDARD NORMAL DISTRIBUTION SECTION 6.2

  2. NORMAL PROBABILITY DISTRIBUTION • The mean, median and mode are the same. • The distribution is bell-shaped and symmetrical around the mean. • The total area under the curve is equal to 1. • The left side and the right side of the normal probability distribution extend indefinitely, never quite touching the horizontal axis.

  3. NORMAL DISTRIBUTION • If a continuous random variable has a distribution with a graph that is symmetric and bell-shaped and it can be described by the equation we say that is has a normal distribution.

  4. STANDARD NORMAL DISTRIBUTION • Is a normal distribution with the parameters of µ = 0 and σ = 1. The total area under its density curve is equal to 1.

  5. FINDING PROBABILITIES • This section will have you find the probability / area under the curve to the z – score. • There are 5 unique style questions • P(a < z < b) • P(a < z ) • P(z < b) • Zα • P%

  6. P(a < z < b) • The probability that the z score is between a and b.

  7. P(a < z) • The probability that the z score is greater than a.

  8. P(z < b) • The probability that the z score is less than b. b

  9. Zα • The z-score with an area of α to its right.

  10. P% • The z-score with an area from 0 to the given %. • Change% to decimal, % = α .

  11. Calculator – Inequality • How to find the probability with a TI • Go to folder DISTR (2nd Vars) • Press 2 (normalcdf) – gives percentage of area under a standard normal distribution. • Normalcdf(lower bound, upper bound) OR • Normalcdf( lower bound, upper bound, mean , standard deviation)

  12. TABLE - Inequality • Given the z score table the graph at the top of the page shows you the probability / area. Understand different z scores shade different sides, so be careful. • The probabilities / areas are on the inside. • The z – score is broken up into two parts • The whole and the tenth number are on the side • The hundredth number is on the top • Match the two together to find the prob / area

  13. EXAMPLE • Find the probability of a normal standard deviation between -0.62 and 1.78

  14. EXAMPLE • Find the probability of a normal standard deviation greater than -1.04

  15. EXAMPLE • Find the probability of a normal standard deviation less than -1.04

  16. Calculator - Zα • Subtract : 1 – α • Distr (2nd vars) • InvNorm( • Type in 1 – α • enter

  17. Table - Zα • Look at the shaded region of the z table • This α connects with a shaded region to the right, if you need the shaded region to the left then subtract from 1, 1 – α • Find the value inside the z score table • Look up and to the left and find the z value

  18. EXAMPLE • Evaluate z0.025

  19. Calculator - P% • Distr button (2nd vars) • Press number 3 invNorm • InvNorm (area → decimal no %)

  20. Table - P% • Look at the middle section of the z score table • Convert & to decimal • Find the probability • If exact probability is not there adjust accordingly • Move up and left for the two part and combine them.

  21. EXAMPLE • Evaluate P90

  22. EXAMPLE • Evaluate z0.025

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