The Normal DistributionBret LargetDepartment of StatisticsUniversity of Wisconsin - MadisonSeptember 28, 2004Statistics 371, Fall 2004The Normal Distribution• The Normal Distribution is the most important distributionof continuous random variables.• The normal density curve is the famous symmetric, bell-shaped curve.• The central limit theorem is the reason that the normalcurve is so important. Essentially, many statistics that wecalculate from large random samples will have approximatenormal distributions (or distributions derived from normaldistributions), even if the distributions of the underlyingvariables are not normally distributed.• This fact is the bas is of most of the m ethods of statisticalinference we will study in the last half of the course.• Chapter 4 introduces the normal distribution as a probabilitydistribution.• Chapter 5 culminates in the central limit theorem, theprimary theoretical justification for most of the methods ofstatistical inference in the remainder of the textbook.Statistics 371, Fall 2004 1The Normal DensityNormal curves have the following bell-shaped, symmetric density.f(y) =1σ√2πe−12y−µσ2Parameters: The parameters of a normal curve are the mean µand the standard deviation σ.Here is an example of a norma l curve with µ = 100 and σ = 20.Normal Distribution mu = 100 , sigma = 20Probability Density50 100 150−3 −2 −1 0 1 2 3Statistics 371, Fall 2004 2Standardization• All normal curves have the same essential shape.• Every normal curve can be drawn in exactly the sam e manner,just by changing labels on the axis.• Thestandard normal curve is the normal curve with meanµ = 0 and standard deviation σ = 1.• Thestandardization formula isZ =Y − µσ• Every problem that asks for an area under a normal curve issolved by firstfinding an equivalent problem for the standardnormal curve.• The justification for this comes from calculus. An area undera normal curve is a definite integral. The integral is simplifiedby using the standardization formula.Statistics 371, Fall 2004 3The 68–95–99.7 RuleFor every normal curve, 68% of the area is within one SD ofthe mean,95% of the area is within two SDs of the mean,and99.7% of the area is within three SDs of the mean.Normal Distribution mu = 0 , sigma = 1Probability Density−4 −2 0 2 4P( −1 < X < 1 ) = 0.6827P( X < −1 ) = 0.1587P( X > 1 ) = 0.1587−3 −2 −1 0 1 2 3Normal Distribution mu = 0 , sigma = 1Probability Density−4 −2 0 2 4P( −2 < X < 2 ) = 0.9545P( X < −2 ) = 0.0228P( X > 2 ) = 0.0228−3 −2 −1 0 1 2 3Normal Distribution mu = 0 , sigma = 1Probability Density−4 −2 0 2 4P( −3 < X < 3 ) = 0.9973P( X < −3 ) = 0.0013P( X > 3 ) = 0.0013−3 −2 −1 0 1 2 3Statistics 371, Fall 2004 4The Standard Normal Table• The standard normal table lists the area to the left of z underthe standard normal curve for each value from − 3. 49 to 3.49by 0.01 increments.• The normal ta ble is on the inside front cover of yourtextbook.• Numbers in the margins represent z.• Numbers in the middle of the table are areas to the left of z.R can do this for general values of z, and R can do thestandardization for you.Statistics 371, Fall 2004 5Example CalculationsSuppose that egg shell thickness is normally distributed with amean of 0.381 mm and a standard deviation of 0.031 mm. Hereare a large number of example calculations.Statistics 371, Fall 2004 6Example Area CalculationArea to the left. Find the proportion of eggs with shellthickness less than 0.34 mm.> gnorm(0.381, 0.031, b = 0.34)Normal Distribution mu = 0.381 , sigma = 0.031Probability Density0.25 0.30 0.35 0.40 0.45 0.50P( X < 0.34 ) = 0.093P( X > 0.34 ) = 0.907−3 −2 −1 0 1 2 3Statistics 371, Fall 2004 7Example Area CalculationArea to the right. Find the proportion of eggs with shellthickness more than 0.36 mm.> gnorm(0.381, 0.031, a = 0.36)Normal Distribution mu = 0.381 , sigma = 0.031Probability Density0.25 0.30 0.35 0.40 0.45 0.50P( X < 0.36 ) = 0.2491P( X > 0.36 ) = 0.7509−3 −2 −1 0 1 2 3Statistics 371, Fall 2004 7Example Area CalculationArea between two values. Find the proportion of eggs withshell thickness between 0.34 and 0. 36 mm .> gnorm(0.381, 0.031, a = 0.34, b = 0.36)Normal Distribution mu = 0.381 , sigma = 0.031Probability Density0.25 0.30 0.35 0.40 0.45 0.50P( 0.34 < X < 0.36 ) = 0.1561P( X < 0.34 ) = 0.093P( X > 0.36 ) = 0.7509−3 −2 −1 0 1 2 3Statistics 371, Fall 2004 7Example Area CalculationArea outside two values. Find the proportion of eggs withshell thickness smaller than 0.32 mm or greater than 0.40 mm.> gnorm(0.381, 0.031, a = 0.32, b = 0.4)Normal Distribution mu = 0.381 , sigma = 0.031Probability Density0.25 0.30 0.35 0.40 0.45 0.50P( 0.32 < X < 0.4 ) = 0.7055P( X < 0.32 ) = 0.0245P( X > 0.4 ) = 0.27−3 −2 −1 0 1 2 3Statistics 371, Fall 2004 7Example Area CalculationCentral area. Find the proportion of eggs with shell thicknesswithin 0.05 mm of the mean.> gnorm(0.381, 0.031, a = 0.381 - 0.05, b = 0.381 + 0.05)Normal Distribution mu = 0.381 , sigma = 0.031Probability Density0.25 0.30 0.35 0.40 0.45 0.50P( 0.331 < X < 0.431 ) = 0.8932P( X < 0.331 ) = 0.0534P( X > 0.431 ) = 0.0534−3 −2 −1 0 1 2 3Statistics 371, Fall 2004 7Example Area CalculationTwo-tail area. Find the proportion of eggs with shell thicknessmore than 0.07 mm from the mean.> gnorm(0.381, 0.031, a = 0.381 - 0.07, b = 0.381 + 0.07)Normal Distribution mu = 0.381 , sigma = 0.031Probability Density0.25 0.30 0.35 0.40 0.45 0.50P( 0.311 < X < 0.451 ) = 0.9761P( X < 0.311 ) = 0.012P( X > 0.451 ) = 0.012−3 −2 −1 0 1 2 3Statistics 371, Fall 2004 7QuantilesQuantile calculations ask you to use the normal table backwards.You know the area but need to find the point or points on thehorizontal axis.Statistics 371, Fall 2004 8Example Quantile CalculationsPercentile. What is the 90th percentile of the egg shellthickness distribution?> gnorm(0.381, 0.031, quantile = 0.9)Normal Distribution mu = 0.381 , sigma = 0.031Probability Density0.25 0.30 0.35 0.40 0.45 0.50z = 1.28P( X < 0.4207 ) = 0.9−3 −2 −1 0 1 2 3Statistics 371, Fall 2004 9Example Quantile CalculationsUpper cut-off point. What value cuts off the top 15% of eggshell thicknesses?> gnorm(0.381, 0.031, quantile = 0.85)Normal Distribution mu = 0.381 , sigma = 0.031Probability Density0.25 0.30 0.35 0.40 0.45 0.50z = 1.04P( X < 0.4131 ) = 0.85−3 −2 −1 0 1 2 3Statistics 371, Fall 2004 9Example Quantile
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