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Normal Distributions

Normal Distributions. 2013. 10. Normal distribution. 정규분포의 예. 두루마리 화장지의 길이 돼지 몸무게 대학교 4 학년 남학생의 키 광장동의 집값 은행의 이자율 등. Properties of Normal Distribution Curve. The normal (distribution) curve

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Normal Distributions

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  1. Normal Distributions 2013. 10

  2. Normal distribution

  3. 정규분포의 예 • 두루마리 화장지의 길이 • 돼지 몸무게 • 대학교 4학년 남학생의 키 • 광장동의 집값 • 은행의 이자율 등

  4. Properties of Normal Distribution Curve • The normal (distribution) curve • From μ–σ to μ+σ: contains about 68% of the measurements (μ: mean, σ: standard deviation) • From μ–2σ to μ+2σ: contains about 95% of it • From μ–3σ to μ+3σ: contains about 99.7% of it

  5. Empirical Rule About 95% of the area lies within 2 standard deviations About 99.7% of the area lies within 3 standard deviations of the mean About 68% of the area lies within 1 standard deviation of the mean 68%

  6. Determining Intervals x 3.3 3.6 3.9 4.2 4.5 4.8 5.1 An instruction manual claims that the assembly time for a product is normally distributed with a mean of 4.2 hours and standard deviation 0.3 hour. Determine the interval in which 95% of the assembly times fall. 95% of the data will fall within 2 standard deviations of the mean. 4.2 – 2 (0.3) = 3.6 and 4.2 + 2 (0.3) = 4.8. 95% of the assembly times will be between 3.6 and 4.8 hrs.

  7. The Standard Score The standard score, or z-score, represents the number of standard deviations a random variable x falls from the mean. The test scores for a civil service exam are normally distributed with a mean of 152 and a standard deviation of 7. Find the standard z-score for a person with a score of: (a) 161 (b) 148 (c) 152 (a) (b) (c)

  8. The Standard Normal Distribution The standard normal distribution has a mean of 0 and a standard deviation of 1. Using z-scores any normal distribution can be transformed into the standard normal distribution. z –4 –3 –2 –1 0 1 2 3 4

  9. Cumulative Areas The total area under the curve is one. z –3 –2 –1 0 1 2 3 • The cumulative area is close to 0 for z-scores close to –3.49. • The cumulative area for z = 0 is 0.5000. • The cumulative area is close to 1 for z-scores close to 3.49.

  10. A Practical Exampleof Normal Dist. • Your company packages sugar in 1 kg bags. • When you weigh a sample of bags you get these results: • 1007g, 1032g, 1002g, 983g, 1004g, ... (a hundred measurements) • Mean = 1010g • Standard Deviation = 20g • Some values are less than 1000g ... can you fix that?

  11. The normal distribution of your measurements looks like this: • 31% of the bags are less than 1000g, which is cheating the customer!

  12. How to reduce “less than 1000g sugar bag” ? • if 1000g was at -3 standard deviations there would be only 0.1% (very small) • if 1000g was at -2.5 standard deviations there would be only 0.6% (small) • How ? • increase the amount of sugar in each bag (this would change the mean), or • make it more accurate (this would reduce the standard deviation)

  13. Solution • First method: adjust the mean amount in each bag • 2.5(standard deviation) * 20g = 50g • Second method: Adjust the accuracy of the machine • we can keep the same mean (of 1010g), but then we need 2.5 standard deviations to be equal to 10g: • 10g / 2.5 = 4g, so the standard deviation should be 4g

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