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Stat350, Lecture#4 :Density curves and normal distribution

Stat350, Lecture#4 :Density curves and normal distribution. Try to draw a smooth curve overlaying the histogram. The curve is a mathematical model for the distribution. It is an idealized description of the data capturing the overall pattern but ignores minor irregularities and outliers.

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Stat350, Lecture#4 :Density curves and normal distribution

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  1. Stat350, Lecture#4 :Density curves and normal distribution • Try to draw a smooth curve overlaying the histogram. • The curve is a mathematical model for the distribution. • It is an idealized description of the data capturing the overall pattern but ignores minor irregularities and outliers.

  2. A density curve is always described by a non-negative function , so lies above the horizontal axis. • The total area under the curve is 1. • The area under the curve for any interval is the proportion of observations falling in that interval. • No set of real data exactly follows a density curve.

  3. Mean and median of a density curve • Median is the point which divides the area under the curve into equal halves. • Mean is a balance point, where the curve will balance if it was made of solid metal • For symmetric distribution, mean and median are same. • For skewed distributions, mean is pulled farther towards the longer tail than the median.

  4. Normal Distribution • Normal distributions : an idealized curve, symmetric, unimodal and bell-shaped • The density function is described by two parameters, the mean of this idealized distribution, µ and the standard deviation ơ . • Remember that xbar and s are always calculated from sample data, the mean and standard distribution of the hypothetical distributions are denoted by the greek symbols.

  5. Locating µ and ơ by eye • µ is the point of symmetry • ơ is the point where the graph changes curvature, the slope gets flatter. • These are special properties of normal distribution, not true for mean and s.d. of any distribution • normal distribution describes many real data • There are many other distributions which can be approximated by a normal distribution for large enough sample size.

  6. The 68-95-99.7 rule • For a normal distribution with mean µ and s.d. , 68% of the observations fall within  of the mean µ, 95% fall within 2 of the mean µ, 99.7% fall within 3 of the mean µ. • Remembering this rule helps us to roughly guess about probabilities regarding normal distributions rather than making constant calculations.

  7. Example: Using the 68-95-99.7 rule • Height of adult men ~ N(69,2.5) • What percent of men are taller than 74 inches? • The 95% rule says the middle 95% have height between 69-5 to 69+5 or between 64 to 74 inches. So the top 2.5% are taller than 74 inches.

  8. Now suppose the height distribution is N(64,2.5). • What percent of men are shorter than 66.5 inches? • The 68% of the rule says the middle 68% have height between 64-2.5 to 64+2.5 or between 61.5 to 66.5, because of symmetry, the top 16% are taller than 66.5 and the lower 16% are shorter than 61.5 inches. • So 84% men are shorter than 66.5 inches.

  9. The standard normal distribution • X~N(,) • The standardized value of x is: z= (x- )/  This is called a z-score. • A z-score tells us how many standard deviations the actual observation is from the mean and which direction. • observations greater than the mean have positive z-score, below the mean have negative z-score.

  10. Example • The amount of rainfall in a month in Indiana follows Normal distribution with mean 4 inches and standard deviation 2.1 inches • A rainfall of 3.3 inches is standardized as: (3.3-4)/2.1 =-.33 • A rainfall of 3.3 inches is .33 standard deviations lower (because of the negative sign) than the mean.

  11. Standard Normal Distribution • If X~N((,) then Z=(X- )/ ~ N(0,1) • If you standardize a variable which has a normal distribution, you get a variable with normal distribution with mean 0 and standard deviation 1. • N(0,1) distribution is called the standard normal distribution. • Table A at the back of the book tabulates area under the standard normal curves.

  12. Using the normal probability table • State the problem in terms of the actual observed variable. • Standardize the X variable to Z • Use the standard normal probability table to find proportion or probability under the standard normal curve. • We will do many normal calculations now on our good old blackboard….

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