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Normal Distributions

Normal Distributions. Symmetrical bell-shaped (unimodal) density curve Above the horizontal axis N( m , s ) The transition points occur at m + s Probability is calculated by finding the area under the curve As s increases , the curve flattens & spreads out

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Normal Distributions

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  1. Normal Distributions • Symmetrical bell-shaped (unimodal) density curve • Above the horizontal axis • N(m, s) • The transition points occur at m+s • Probability is calculated by finding the area under the curve • As sincreases, the curve flattens & spreads out • As sdecreases, the curve gets taller and thinner How is this done mathematically?

  2. Normal distributions occur frequently. • Length of newborn child • Height • Weight • ACT or SAT scores • Intelligence • Number of typing errors • Chemical processes

  3. s s A B 6 Do these two normal curves have the same mean? If so, what is it? Which normal curve has a standard deviation of 3? Which normal curve has a standard deviation of 1? YES B A

  4. Empirical Rule • Approximately 68% of the observations fall within s of m • Approximately 95% of the observations fall within 2s of m • Approximately 99.7% of the observations fall within 3s of m

  5. 68% 71 Suppose that the height of male students at NSHS is normally distributed with a mean of 71 inches and standard deviation of 2.5 inches. What is the probability that the height of a randomly selected male student is more than 73.5 inches? 1 - .68 = .32 P(X > 73.5) = 0.16

  6. Standard Normal Density Curves Always has m = 0 & s = 1 To standardize: Must have this memorized!

  7. Strategies for finding probabilities or proportions in normal distributions • State the probability statement • Draw a picture • Calculate the z-score • Use Normalcdf on my calculator

  8. Will my calculator do any of this normal stuff? • Normalpdf – use for graphing ONLY • Normalcdf – will find probability of area from lower bound to upper bound • Invnorm (inverse normal) – will find z-score for probability

  9. The lifetime of a certain type of battery is normally distributed with a mean of 200 hours and a standard deviation of 15 hours. What proportion of these batteries can be expected to last less than 220 hours? Write the probability statement Draw & shade the curve P(X < 220) = .9082 Look up z-score in table Calculate z-score

  10. The lifetime of a certain type of battery is normally distributed with a mean of 200 hours and a standard deviation of 15 hours. What proportion of these batteries can be expected to last more than 220 hours? P(X>220) = 1 - .9082 = .0918

  11. The lifetime of a certain type of battery is normally distributed with a mean of 200 hours and a standard deviation of 15 hours. How long must a battery last to be in the top 5%? Look up in table 0.95 to find z- score P(X > ?) = .05 .95 .05 1.645

  12. The heights of the female students at NSHS are normally distributed with a mean of 65 inches. What is the standard deviation of this distribution if 18.5% of the female students are shorter than 63 inches? What is the z-score for the 63? P(X < 63) = .185 -0.9 63

  13. The heights of female teachers at NSHS are normally distributed with mean of 65.5 inches and standard deviation of 2.25 inches. The heights of male teachers are normally distributed with mean of 70 inches and standard deviation of 2.5 inches. • Describe the distribution of differences of heights (male – female) teachers. Normal distribution with m = 4.5 & s = 3.3634

  14. 4.5 • What is the probability that a randomly selected male teacher is shorter than a randomly selected female teacher? P(X<0) = .0901

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