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Download The notation used to describe the normal distribution
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Normal Distribution When the random variable, X, is allowed to be continuous, one of the possible types of distributions it can have is the Normal distribution. It is bell‐shaped and symmetric about the mean. μ - pronounced "mu" - mean σ - pronounced "sigma" -standard deviation The notation used to describe the normal distribution, of the random variable X, is X ~ N(μ,σ2) where μ is the mean and σ2 is the variance. The scale on the y‐axis is rarely labeled. Usually it represents the relative frequency (i.e. probability). No situation in the real world is perfect, but many natural relationships, when displayed as a histogram, will form a normal distribution. 1 Ex. Consider the normal distribution N(13, 16). a) Determine the mean. b) Determine the standard deviation. c) What interval does 68% of the data fall between? d) What interval does 99.7% of the data fall between? Ex. An engineer wants to build a car seat that 95% of the adult population can use. the average adult in North America has a mass of 158.3lbs. with a standard deviation of 30lbs. a) What range of masses would include 95% of the general population? 95% of the adult population in North America is between 98.3 lbs and 218.3 lbs. b) What assumption has been made? Is it valid? The assumption is that masses for adults in North America are normally distributed. This is a valid assumption since this is a natural distribution. 2 Area Under the Probability Normal Curve The area under EVERY probability normal curve is one (1). Therefore the percent of the data that lies between two values in a normal distribution is equivalent to the area under the curve between these two values. Area under normal = % of data that lies curve between two between two points points (a and b) (a and b) Ex. Given X ~ N(40,22), draw a diagram that represents the percent of data that have the following values for x. a. b. Ex. The weights of 10 000 females athletes are normally distributed. The mean weight is 55kg and the standard deviation is 5kg. a. find the number of athletes weighing between 50 and 60kg. 50 to 60 kg represents µ ± σ, which is 68% of the female athletes. To determine the number, we take 68% of 10000. Therefore, 6 800 athletes fall into this range. b. find the number of athletes weighing less than 45kg. range: ≤45kg 45 kg occurs at µ ‐ 2σ. The % in the highlighted region is given by: (100% - 95%)÷2 = 2.5% (think about this) Therefore, 10000*0.025 = 250 athletes fall into this range. 3 Homework: Pg 176 #1, 3iii, 6, 8, 9, 12, 14 4 Attachments MDM4U 1.3 Olympics Mens Field Trend.ftm MDM4U 1.4 Elements.ftm MDM4U 1.4 Curve of Best Fit.ftm MDM4U 1.5.ppt MDM4U 2.45 Survey.doc MDM4U 2.45 Public Opinion Survey on Youth and Sports.doc
