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Exploring Probability and Statistics with Normal Distributions

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liuz
16 diapositives · 280 Views · Sep 27, 2014

Learn to calculate probabilities using normal distributions, understand key aspects of a normal curve, and solve examples involving pricing and weights. This lesson covers mean, median, mode, range, interquartile range, creating box-and-whisker plots, and standard deviation calculations.

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Exploring Probability and Statistics with Normal Distributions
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  1. Unit 4 – Probability and Statistics Section 12.7 Day 10

  2. Warm-Up Using the given data set: 77, 76, 83, 88, 90, 85, 81, 77, 95, 89 • Find the mean, median, mode • Find the range and interquartile range • Create a box-and-whisker plot of the data • Find the standard deviation (Show all your work.)

  3. Warm-Up Answers Data set in order: 76, 77, 77, 81, 83, 85, 88, 89, 90, 95 • Mean = 84.1; Median = 84 ; Mode = 77 • Range = 19; Interquartile range = 12 • Create a box-and-whisker plot of the data • Standard deviation σ = 6.1

  4. Section 12.7 Normal Distributions Goal: calculate probabilities using normal distributions Normal Curve – symmetrical bell-shaped curve Important symbols • σ (Sigma) represents standard deviation • x (x-bar) represents mean

  5. Section 12.7 Normal Distributions Important Aspects of a Normal Curve • Total Area under the curve is 1 • 68% of the area lies within 1 standard deviation of the mean • 95% of the area lies within 2 standard deviations of the mean • 99.7% of the area lies within 3 standard deviations of the mean

  6. Section 12.7 Normal Distributions 34% 34% 13.5% 13.5% 0.15% 0.15% 2.35% 2.35% -3σ -2σ -σ x +σ +2σ +3σ

  7. Example 1 The price of sandals is normally distributed with a mean of $36 and a standard deviation of $9. • What percent were priced between $18 and $72? • What is the probability that a randomly chosen pair is priced between $9 and $18?

  8. Example 1 (cont.) 34% 34% 13.5% 13.5% 0.15% 0.15% 2.35% 2.35% $9 $18 $27 $36 $45 $54 $63 A) What percent were priced between $18 and $72? 97.5%

  9. Example 1 (cont.) 34% 34% 13.5% 13.5% 0.15% 0.15% 2.35% 2.35% $9 $18 $27 $36 $45 $54 $63 B) What is the probability that a randomly chosen pair is priced between $9 and $18? 2.35%

  10. Example 2 The weight of strawberries packed in a medium-sized container was normally distributed with a mean of 16.18 ounces and a standard deviation of 0.34 ounce. • If you randomly choose 2 containers of strawberries, what is the probability that both weigh less than 15.5 ounces? • If you randomly choose 2 containers of strawberries, What is the probability that both of the containers weighs at least 15.84 ounces? • If you randomly choose 2 containers of strawberries, that one container weighs between 15.5 and 15.84 ounces, and the 2nd container weighs at least 16.86 ounces?

  11. Example 2 (cont.) 34% 34% 13.5% 13.5% 0.15% 0.15% 2.35% 2.35% (oz.) 15.16 15.5 15.84 16.18 16.52 16.86 17.2 A) Both containers weigh less than 15.5 oz? 0.025 0.025 = 0.000625 or 0.0625%

  12. Example 2 (cont.) 34% 34% 13.5% 13.5% 0.15% 0.15% 2.35% 2.35% (oz.) 15.16 15.5 15.84 16.18 16.52 16.86 17.2 B) Both containers weigh at least 15.84 oz? 0.84 0.84 = 0.7056 or 70.56%

  13. Example 2 (cont.) 34% 34% 13.5% 13.5% 0.15% 0.15% 2.35% 2.35% (oz.) 15.16 15.5 15.84 16.18 16.52 16.86 17.2 C) Between 15.5 and 15.84 oz and at least 16.86? 0.135 (0.0235 + .0015) = 0.003375 or 0.3375%

  14. PRACTICE P. 749-750 #19 - 28

  15. PRACTICE ANSWERS 19) 50% 20) 15.85% 21) 2.5% 22) 27% 23) 0.68 24) 0.4985 25) 0.9735 26) 0.84 27) 0.84 28) 0.975

  16. HOMEWORK P. 750-751 #29 – 31, 38 - 49