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chAPTER five

chAPTER five. Normal Probability Distributions. Section 5.1. Intro to Normal Distributions & the STANDARD Normal Distribution. Properties of a Normal Distribution. 1. mean, median, and mode are equal 2. bell shaped and symmetric about the mean 3. total area under the curve is 1

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chAPTER five

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  1. chAPTER five Normal Probability Distributions

  2. Section 5.1 Intro to Normal Distributions & the STANDARD Normal Distribution

  3. Properties of a Normal Distribution • 1. mean, median, and mode are equal • 2. bell shaped and symmetric about the mean • 3. total area under the curve is 1 • 4. the curve approaches, but never touches the x axis as it extends away from the mean

  4. The Standard Normal Distribution • A Normal distribution with • mean = 0 and • standard deviation = 1

  5. Properties of Standard Normal Distribution • The cumulative area is close to 0 for z-scores close to z = -3.49 • The cumulative area increases as the z-score increases. • The cumulative area for z = 0 is 0.5000 • The cumulative area is close to 1 for z-scores close to z = 3.49

  6. Table #4 (Starts on p A16) • Sketch the curve and shade the appropriate area under the curve. • To find the area LEFT of z, find the area that corresponds to z on the table. • To find the area RIGHT of z, find the area that corresponds to z on the table, then subtract from 1. • To find the are BETWEEN 2 z-scores, find each area, then subtract.

  7. EX: Find the area for each: • 1. left of z = -1.02 • 2. right of z = 2.30 • 3. left of z = 0.45 • 4. right of z = - 1.99 • 5. between z = -2.34 and z = 1.89

  8. Section 5.2 Normal Distributions: Finding Probabilities

  9. Area under curve => Probability • 1. Sketch a normal curve. • 2. Find the z-score Recall: z = x - µ σ • 3. Plot your z-score and shade as indicated. • 4. Find the area under the curve using the standard normal distribution table (table #4) • 5. Interpret the results to answer the question.

  10. EX: assume normal distribution with µ = 174 and δ = 20 Find P(x < 200) Find P(x > 155) Find P(172 < x < 192)

  11. 13. A survey was conducted to measure the heights of US men. In the survey, respondents were grouped by age. In the 20-29 age group, the heights were normally distributed, with a mean of 69.9 inches and a standard deviation of 3.0 inches. A study participant is randomly selected. Find each prob: • A) his height is less than 66 in. • B) his height is between 66 and 72 in. • C) his height is more than 72 in.

  12. 20. The times per workout an athlete uses a stairclimber are normally distributed, with a mean of 20 minutes and a standard deviation of 5 minutes. An athlete is randomly selected. Find each probability. • A) the athlete uses a stairclimber for less than 17 minutes. • B) the athlete uses a stairclimber between 20 and 28 minutes. • C) the athlete uses a stairclimber for more than 30 minutes.

  13. 26. Monthly utility bills are normally distributed with a mean of $100 and a standard deviation of $12. • A) What percent of the utility bills are more than $125? • B) If 300 utility bills are randomly selected, about how many would you expect to be less than $90?

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