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Normal Distribution

Normal Distribution. Practice with z-scores. Probabilities are depicted by areas under the curve. Total area under the curve is 1 Only have a probability from width For an infinite number of z scores each point has a probability of 0 (for the single point)

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Normal Distribution

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  1. Normal Distribution Practice with z-scores

  2. Probabilities are depicted by areas under the curve • Total area under the curve is 1 • Only have a probability from width • For an infinite number of z scores each point has a probability of 0 (for the single point) • The area in red is equal to p(z > 1) • The area in blue is equal to p(-1< z <0) • Since the properties of the normal distribution are known, areas can be looked up on tables or calculated on computer.

  3. Strategies for finding probabilities for the standard normal random variable. • Draw a picture of standard normal distribution depicting the area of interest • Look up the areas using the table • Do the necessary addition and subtraction

  4. Find p(0<Z<1.23) • In your Howell appendix just note the ‘Mean to Z’ column • .391

  5. Find p(-1.57<Z<0) • Same thing here, but note your table doesn’t distinguish between positive and negative • As it is a symmetric curve, the probability is the same either way • .442

  6. Calculate p(-1.2<Z<.78) • Here we just find the ‘Mean to z’ for .78, and then for 1.2, and just add them together • .667

  7. Find p(Z>.78) • This is more the style of probability we’ll be concerned with primarily • What’s the likelihood of getting this score, or more extreme? • .218

  8. Example: IQ • A common example is IQ • IQ scores are theoretically normally distributed. • Mean of 100 • Standard deviation of 15

  9. Example IQ • What’s the probability of getting a score between 100 and 115 IQ?

  10. Work time... • What is the area for scores less than: z = -2.5? • What is the area between z =1.5 and 2.0? • What z score cuts off the highest 10% of the distribution? • What two z scores enclose the middle 50% of the distribution? • On your own: • If 500 scores are normally distributed with mean = 50 and SD = 10, and an investigator throws out the 20 most extreme scores (10 high and 10 low), what are the approximate highest and lowest scores that are retained?

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