Ch06 Normal Distribution and Other Continuous Distributions

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Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc. Chap 6- 1 Chapter 6 The Normal Distribution and Other Continuous Distributions Basic Business Statistics 11 th Edition

Learning Objectives:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 2 Learning Objectives In this chapter, you learn: To compute probabilities from the normal distribution To use the normal probability plot to determine whether a set of data is approximately normally distributed To compute probabilities from the uniform distribution To compute probabilities from the exponential distribution To compute probabilities from the normal distribution to approximate probabilities from the binomial distribution

Continuous Probability Distributions:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 3 Continuous Probability Distributions A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values) thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value depending only on the ability to precisely and accurately measure

The Normal Distribution:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 4 The Normal Distribution ‘ Bell Shaped ’ Symmetrical Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: +  to   Mean = Median = Mode X f(X) μ σ

The Normal Distribution Density Function:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 5 The Normal Distribution Density Function The formula for the normal probability density function is Where e = the mathematical constant approximated by 2.71828 π = the mathematical constant approximated by 3.14159 μ = the population mean σ = the population standard deviation X = any value of the continuous variable

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Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 6 By varying the parameters μ and σ , we obtain different normal distributions Many Normal Distributions

The Normal Distribution Shape:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 7 The Normal Distribution Shape X f(X) μ σ Changing μ shifts the distribution left or right . Changing σ increases or decreases the spread.

The Standardized Normal:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 8 The Standardized Normal Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardized normal distribution (Z) Need to transform X units into Z units The standardized normal distribution ( Z ) has a mean of 0 and a standard deviation of 1

Translation to the Standardized Normal Distribution:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 9 Translation to the Standardized Normal Distribution Translate from X to the standardized normal (the “Z” distribution) by subtracting the mean of X and dividing by its standard deviation : The Z distribution always has mean = 0 and standard deviation = 1

The Standardized Normal Probability Density Function:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 10 The Standardized Normal Probability Density Function The formula for the standardized normal probability density function is Where e = the mathematical constant approximated by 2.71828 π = the mathematical constant approximated by 3.14159 Z = any value of the standardized normal distribution

The Standardized Normal Distribution:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 11 The Standardized Normal Distribution Also known as the “Z” distribution Mean is 0 Standard Deviation is 1 Z f(Z) 0 1 Values above the mean have positive Z-values, values below the mean have negative Z-values

Example:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 12 Example If X is distributed normally with mean of 100 and standard deviation of 50 , the Z value for X = 200 is This says that X = 200 is two standard deviations (2 increments of 50 units) above the mean of 100.

Comparing X and Z units:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 13 Comparing X and Z units Z 100 2.0 0 200 X Note that the shape of the distribution is the same, only the scale has changed. We can express the problem in original units (X) or in standardized units (Z) ( μ = 100, σ = 50) ( μ = 0, σ = 1)

Finding Normal Probabilities :

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 14 Finding Normal Probabilities a b X f(X) P a X b ( ) ≤ Probability is measured by the area under the curve ≤ P a X b ( ) < < = (Note that the probability of any individual value is zero)

Probability as Area Under the Curve:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 15 f(X) X μ Probability as Area Under the Curve 0.5 0.5 The total area under the curve is 1.0 , and the curve is symmetric, so half is above the mean, half is below

The Standardized Normal Table:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 16 The Standardized Normal Table The Cumulative Standardized Normal table in the textbook (Appendix table E.2) gives the probability less than a desired value of Z (i.e., from negative infinity to Z) Z 0 2.00 0.9772 Example: P(Z < 2.00) = 0.9772

The Standardized Normal Table:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 17 The Standardized Normal Table The value within the table gives the probability from Z =   up to the desired Z value .9772 2.0 P(Z < 2.00) = 0.9772 The row shows the value of Z to the first decimal point The column gives the value of Z to the second decimal point 2.0 . . . (continued) Z 0.00 0.01 0.02 … 0.0 0.1

General Procedure for Finding Normal Probabilities:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 18 General Procedure for Finding Normal Probabilities Draw the normal curve for the problem in terms of X Translate X-values to Z-values Use the Standardized Normal Table To find P(a < X < b) when X is distributed normally:

Finding Normal Probabilities:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 19 Finding Normal Probabilities Let X represent the time it takes to download an image file from the internet. Suppose X is normal with mean 8.0 and standard deviation 5.0. Find P(X < 8.6) X 8.6 8.0

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Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 20 Let X represent the time it takes to download an image file from the internet. Suppose X is normal with mean 8.0 and standard deviation 5.0. Find P(X < 8.6) Z 0.12 0 X 8.6 8 μ = 8 σ = 10 μ = 0 σ = 1 (continued) Finding Normal Probabilities P(X < 8.6) P(Z < 0.12)

Solution: Finding P(Z < 0.12):

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 21 Z 0.12 Z .00 .01 0.0 .5000 .5040 .5080 .5398 .5438 0.2 .5793 .5832 .5871 0.3 .6179 .6217 .6255 Solution: Finding P(Z < 0.12) .5478 .02 0.1 . 5478 Standardized Normal Probability Table (Portion) 0.00 = P( Z < 0.12) P( X < 8.6)

Finding Normal Upper Tail Probabilities:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 22 Finding Normal Upper Tail Probabilities Suppose X is normal with mean 8.0 and standard deviation 5.0. Now Find P(X > 8.6) X 8.6 8.0

Finding Normal Upper Tail Probabilities:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 23 Now Find P(X > 8.6)… (continued) Z 0.12 0 Z 0.12 0.5478 0 1.000 1.0 - 0.5478 = 0.4522 P(X > 8.6) = P(Z > 0.12) = 1.0 - P(Z ≤ 0.12) = 1.0 - 0.5478 = 0.4522 Finding Normal Upper Tail Probabilities

Finding a Normal Probability Between Two Values:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 24 Finding a Normal Probability Between Two Values Suppose X is normal with mean 8.0 and standard deviation 5.0. Find P(8 < X < 8.6) P( 8 < X < 8.6) = P( 0 < Z < 0.12) Z 0.12 0 X 8.6 8 Calculate Z-values:

Solution: Finding P(0 < Z < 0.12):

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 25 Z 0.12 Solution: Finding P(0 < Z < 0.12) 0.0478 0.00 = P( 0 < Z < 0.12) P( 8 < X < 8.6) = P( Z < 0.12) – P(Z ≤ 0) = 0.5478 - .5000 = 0.0478 0.5000 Z .00 .01 0.0 . 5000 .5040 .5080 .5398 .5438 0.2 .5793 .5832 .5871 0.3 .6179 .6217 .6255 .02 0.1 . 5478 Standardized Normal Probability Table (Portion)

Probabilities in the Lower Tail :

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 26 Suppose X is normal with mean 8.0 and standard deviation 5.0. Now Find P(7.4 < X < 8) X 7.4 8.0 Probabilities in the Lower Tail

Probabilities in the Lower Tail :

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 27 Probabilities in the Lower Tail Now Find P(7.4 < X < 8)… X 7.4 8.0 P(7.4 < X < 8) = P(-0.12 < Z < 0) = P(Z < 0) – P(Z ≤ -0.12) = 0.5000 - 0.4522 = 0.0478 (continued) 0.0478 0.4522 Z -0.12 0 The Normal distribution is symmetric, so this probability is the same as P(0 < Z < 0.12)

Empirical Rules:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 28 Empirical Rules μ ± 1 σ encloses about 68.26% of X’s  f(X) X μ μ +1 σ μ -1 σ What can we say about the distribution of values around the mean? For any normal distribution: σ σ 68.26%

The Empirical Rule:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 29 The Empirical Rule μ ± 2 σ covers about 95% of X’s μ ± 3 σ covers about 99.7% of X’s x μ 2 σ 2 σ x μ 3 σ 3 σ 95.44% 99.73% (continued)

Given a Normal Probability Find the X Value:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 30 Steps to find the X value for a known probability: 1. Find the Z value for the known probability 2. Convert to X units using the formula: Given a Normal Probability Find the X Value

Finding the X value for a Known Probability:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 31 Finding the X value for a Known Probability Example: Let X represent the time it takes (in seconds) to download an image file from the internet. Suppose X is normal with mean 8.0 and standard deviation 5.0 Find X such that 20% of download times are less than X. X ? 8.0 0.2000 Z ? 0 (continued)

Find the Z value for 20% in the Lower Tail:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 32 Find the Z value for 20% in the Lower Tail 20% area in the lower tail is consistent with a Z value of -0.84 Z .03 -0.9 .1762 .1736 .2033 -0.7 .2327 .2296 .04 -0.8 . 2005 Standardized Normal Probability Table (Portion) .05 .1711 .1977 .2266 … … … … X ? 8.0 0.2000 Z -0.84 0 1. Find the Z value for the known probability

Finding the X value:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 33 2. Convert to X units using the formula: Finding the X value So 20% of the values from a distribution with mean 8.0 and standard deviation 5.0 are less than 3.80

Evaluating Normality:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 34 Evaluating Normality Not all continuous distributions are normal It is important to evaluate how well the data set is approximated by a normal distribution. Normally distributed data should approximate the theoretical normal distribution: The normal distribution is bell shaped (symmetrical) where the mean is equal to the median. The empirical rule applies to the normal distribution. The interquartile range of a normal distribution is 1.33 standard deviations.

Evaluating Normality:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 35 Evaluating Normality Comparing data characteristics to theoretical properties Construct charts or graphs For small- or moderate-sized data sets, construct a stem-and-leaf display or a boxplot to check for symmetry For large data sets, does the histogram or polygon appear bell-shaped? Compute descriptive summary measures Do the mean, median and mode have similar values? Is the interquartile range approximately 1.33 σ ? Is the range approximately 6 σ ? (continued)

Evaluating Normality:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 36 Evaluating Normality Comparing data characteristics to theoretical properties Observe the distribution of the data set Do approximately 2/3 of the observations lie within mean ± 1 standard deviation? Do approximately 80% of the observations lie within mean ± 1.28 standard deviations? Do approximately 95% of the observations lie within mean ± 2 standard deviations? Evaluate normal probability plot Is the normal probability plot approximately linear (i.e. a straight line) with positive slope? (continued)

Constructing A Normal Probability Plot:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 37 Constructing A Normal Probability Plot Normal probability plot Arrange data into ordered array Find corresponding standardized normal quantile values (Z) Plot the pairs of points with observed data values (X) on the vertical axis and the standardized normal quantile values (Z) on the horizontal axis Evaluate the plot for evidence of linearity

The Normal Probability Plot Interpretation:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 38 A normal probability plot for data from a normal distribution will be approximately linear : 30 60 90 -2 -1 0 1 2 Z X The Normal Probability Plot Interpretation

Normal Probability Plot Interpretation:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 39 Normal Probability Plot Interpretation Left-Skewed Right-Skewed Rectangular 30 60 90 -2 -1 0 1 2 Z X (continued) 30 60 90 -2 -1 0 1 2 Z X 30 60 90 -2 -1 0 1 2 Z X Nonlinear plots indicate a deviation from normality

Evaluating Normality An Example: Mutual Funds Returns:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 40 Evaluating Normality An Example: Mutual Funds Returns The boxplot appears reasonably symmetric, with three lower outliers at -9.0, -8.0, and -8.0 and one upper outlier at 35.0. (The normal distribution is symmetric.)

Evaluating Normality An Example: Mutual Funds Returns:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 41 Evaluating Normality An Example: Mutual Funds Returns Descriptive Statistics (continued) The mean (12.5142) is slightly less than the median (13.1). (In a normal distribution the mean and median are equal.) The interquartile range of 9.2 is approximately 1.46 standard deviations. (In a normal distribution the interquartile range is 1.33 standard deviations.) The range of 44 is equal to 6.99 standard deviations. (In a normal distribution the range is 6 standard deviations.) 72.2% of the observations are within 1 standard deviation of the mean. (In a normal distribution this percentage is 68.26%. 87% of the observations are within 1.28 standard deviations of the mean. (In a normal distribution percentage is 80%.)

Evaluating Normality An Example: Mutual Funds Returns:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 42 Evaluating Normality An Example: Mutual Funds Returns (continued) Plot is approximately a straight line except for a few outliers at the low end and the high end.

Evaluating Normality An Example: Mutual Funds Returns:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 43 Evaluating Normality An Example: Mutual Funds Returns Conclusions The returns are slightly left-skewed The returns have more values concentrated around the mean than expected The range is larger than expected (caused by one outlier at 35.0) Normal probability plot is reasonably straight line Overall, this data set does not greatly differ from the theoretical properties of the normal distribution (continued)

The Uniform Distribution:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 44 The Uniform Distribution The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable Also called a rectangular distribution

The Uniform Distribution:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 45 The Continuous Uniform Distribution: where f(X) = value of the density function at any X value a = minimum value of X b = maximum value of X The Uniform Distribution (continued) f(X) =

Properties of the Uniform Distribution:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 46 Properties of the Uniform Distribution The mean of a uniform distribution is The standard deviation is

Uniform Distribution Example:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 47 Uniform Distribution Example Example: Uniform probability distribution over the range 2 ≤ X ≤ 6: 2 6 0.25 f(X) = = 0.25 for 2 ≤ X ≤ 6 6 - 2 1 X f(X)

Uniform Distribution Example:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 48 Uniform Distribution Example Example: Using the uniform probability distribution to find P(3 ≤ X ≤ 5): 2 6 0.25 P( 3 ≤ X ≤ 5 ) = (Base)(Height) = (2)(0.25) = 0.5 X f(X) (continued) 3 5 4

The Exponential Distribution:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 49 The Exponential Distribution Often used to model the length of time between two occurrences of an event (the time between arrivals) Examples: Time between trucks arriving at an unloading dock Time between transactions at an ATM Machine Time between phone calls to the main operator

The Exponential Distribution:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 50 The Exponential Distribution Defined by a single parameter, its mean λ (lambda) The probability that an arrival time is less than some specified time X is where e = mathematical constant approximated by 2.71828 λ = the population mean number of arrivals per unit X = any value of the continuous variable where 0 < X <

Exponential Distribution Example:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 51 Exponential Distribution Example Example: Customers arrive at the service counter at the rate of 15 per hour. What is the probability that the arrival time between consecutive customers is less than three minutes? The mean number of arrivals per hour is 15, so λ = 15 Three minutes is 0.05 hours P(arrival time < .05) = 1 – e - λ X = 1 – e -(15)(0.05) = 0.5276 So there is a 52.76% probability that the arrival time between successive customers is less than three minutes

The Exponential Distribution In Excel:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 52 The Exponential Distribution In Excel Calculating the probability that an exponential distribution with an mean of 20 is less than 0.1

Normal Approximation to the Binomial Distribution:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 53 Normal Approximation to the Binomial Distribution The binomial distribution is a discrete distribution, but the normal is continuous To use the normal to approximate the binomial, accuracy is improved if you use a correction for continuity adjustment Example : X is discrete in a binomial distribution, so P(X = 4) can be approximated with a continuous normal distribution by finding P(3.5 < X < 4.5)

Normal Approximation to the Binomial Distribution:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 54 Normal Approximation to the Binomial Distribution The closer π is to 0.5, the better the normal approximation to the binomial The larger the sample size n, the better the normal approximation to the binomial General rule: The normal distribution can be used to approximate the binomial distribution if n π ≥ 5 and n(1 – π ) ≥ 5 (continued)

Normal Approximation to the Binomial Distribution:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 55 Normal Approximation to the Binomial Distribution The mean and standard deviation of the binomial distribution are μ = n π Transform binomial to normal using the formula: (continued)

Using the Normal Approximation to the Binomial Distribution:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 56 Using the Normal Approximation to the Binomial Distribution If n = 1000 and π = 0.2, what is P(X ≤ 180)? Approximate P(X ≤ 180) using a continuity correction adjustment: P(X ≤ 180.5) Transform to standardized normal: So P(Z ≤ -1.54) = 0.0618 X 180.5 200 -1.54 0 Z

Chapter Summary:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 6- 57 Chapter Summary Presented key continuous distributions normal, uniform, exponential Found probabilities using formulas and tables Recognized when to apply different distributions Applied distributions to decision problems

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