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Lecture

# Chekyshev’s rule, normal distribution, how to calculate probabilities

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School
Department
Statistics
Course
STAT 301
Professor
Brett Hunter
Semester
Fall

Description
7 September Chekyshev’s Rule For any population distribution, we know that the percentage of observations within k standard deviations of the mean is at least 1 100 ( 1 - k2 ) % For any k ≥ 1 Example We have an exam where mean = 83 and standard deviation = 7. Between what two values do at least 75% of the exam scores fall? 1 2 100 ( 1 - k ) = 75 K = 4 K = 2 At least 75% of the exam scores fall within 2 standard deviations of the mean. Between μ – kσ and μ + kσ 83 – 2(7) = 69 83 + 2(7) = 97 So at least 75% of the exam scores are between a 69 and a 97. Normal Distribution (Gaussian) 2 We denote a random variable X that follows a normal distribution as X ~N (μ, σ ) The normal distribution is defined by the density curve 1 −1 (x−μ) F(x) = 2 (2σ2 √ 2πσ
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