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8-5 - 8-5 Normal Distributions continuous distribution...

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8-5 Normal Distributions continuous distribution found in many application areas e.g. SAT scores, people’s heights, leaf lengths all are distributed according to the normal distribution also called the Gaussian Distribution it is the famous "bell-shaped" curve: distributions have a mean μ and a standard deviation σ the above Normal has μ = 430 and σ = 100 For any Normal Distribution with mean μ and standard deviation σ : 68% of all scores lie within ± 1 σ of the mean 95% are within ± 2 σ 99.7% (virtually all) are within ± 3 σ 8-5 p. 1
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Computing areas (probabilities) under normal distributions Question: What % of the people that take the SAT make scores between 330 and 530? Analysis: mean μ = 430 s.d. σ = 100 picture: 330 = μ-σ 530 = μ + σ % of scores between μ-σ and μ + σ for any normal distribution? We know that one; it's 68% !!! 8-5 p. 2
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Question: What % of the people that take the SAT make scores between 430 and 600? Analysis: mean μ = 430 s.d. σ = 100 600 = how many σ from the mean? z = 100 430 600 - = 1.7 σ We only know percentages for within ± 1 σ, ± 2 σ and ± 3 σ of the mean. Q: Can we find the area under a normal from μ to μ + 1.7 σ A: Yes, we can look it up in a table. Here's the picture: z = 1.7 σ The table on page 723: gives us area ( for any normal distribution !!!) from μ to μ + (any number of σ ) look up 1.7 .4554 Answer : 45.56% 8-5 p. 3
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Z-statistics by computing the number of σ 's a score is from the mean you are computing a number known as: a z-statistic , or a standardized score the unit of measurement of a z-statistic is always standard deviations Example: if μ = 10 and σ = 7, what is the z-statistic
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