Chapter 6

The Normal Distribution and Other Continuous Distributions

by Try Sothearith trysothearith@gmail.com sothearith_t@yahoo.com Tel: 012 585 865 / 016555507

Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.

Learning ObjectivesIn this chapter, you learn: To compute probabilities from the normal distributionTo use the normal probability plot to determine whether a set of data is approximately normally distributedTo 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

Probability DistributionsContinuous Probability DistributionsBinomialHypergeometricPoissonProbability DistributionsDiscrete Probability DistributionsNormalUniformExponentialCh. 5Ch. 6

Continuous Probability DistributionsA continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values)thickness of an itemtime required to complete a tasktemperature of a solutionheight, in inches

These can potentially take on any value, depending only on the ability to measure accurately.

The Normal DistributionProbability DistributionsNormalUniformExponentialContinuous Probability Distributions

The Normal Distribution Bell Shaped Symmetrical Mean, Median and Mode are EqualLocation is determined by the mean, Spread is determined by the standard deviation,

The random variable has an infinite theoretical range: + to Mean = Median = ModeXf(X)

By varying the parameters and , we obtain different normal distributionsMany Normal Distributions

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The Normal Distribution ShapeXf(X)Changing shifts the distribution left or right.Changing increases or decreases the spread.

The Normal Probability Density FunctionThe formula for the normal probability density function isWheree = the mathematical constant approximated by 2.71828 = the mathematical constant approximated by 3.14159 = the population mean = the population standard deviationX = any value of the continuous variable

The Standardized NormalAny 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

Translation to the Standardized Normal DistributionTranslate 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 FunctionThe formula for the standardized normal probability density function isWheree = the mathematical constant approximated by 2.71828 = the mathematical constant approximated by 3.14159Z = any value of the standardized normal distribution

The Standardized Normal DistributionAlso known as the Z distributionMean is 0Standard Deviation is 1Zf(Z)01Values above the mean have positive Z-values, values below the mean have negative Z-values

ExampleIf 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 unitsZ1002.00200XNote that the distribution is the same, only the scale has changed. We can express the problem in original units (X) or in standardized units (Z)( = 100, = 50)( = 0, = 1)

Finding Normal Probabilities Probability is the area under the curve!abXf(X)PaXb()Probability is measured by the area under the curvePaXb()

f(X)XProbability as Area Under the Curve0.50.5The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below

Empirical Rules 1 encloses about 68% of Xsf(X)X+1-1What can we say about the distribution of values around the mean? There are some general rules:68.26%

The Empirical Rule 2 covers about 95% of Xs 3 covers about 99.7% of Xsx22x3395.44%99.73%(continued)

The Standardized Normal Table The Cumulative Standardized Normal table in the textbook (Appendix table E.2) gives the probability less than a desired value for Z (i.e., from negative infinity to Z)

Z02.000.9772Example: P(Z < 2.00) = 0.9772

The Standardized Normal Table The value within the table gives the probability from Z = up to the desired Z value.97722.0P(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 point2.0...(continued) Z 0.00 0.01 0.02

0.00.1

General Procedure for Finding Probabilities Draw the normal curve for the problem in terms of X

Translate X-values to Z-values

Use the Standardized Normal TableTo find P(a < X < b) when X is distributed normally:

Finding Normal ProbabilitiesSuppose X is normal with mean 8.0 and standard deviation 5.0Find P(X < 8.6)X8.68.0

Suppose X is normal with mean 8.0 and standard deviation 5.0. Find P(X < 8.6)

Z0.12 0X8.6 8 = 8 = 10 = 0 = 1(continued)Finding Normal ProbabilitiesP(X < 8.6)P(Z < 0.12)

Z0.12Z.00.010.0.5000.5040.5080.5398.54380.2.5793.5832.58710.3.6179.6217.6255Solution: Finding P(Z < 0.12).5478.020.1.5478Standardized Normal Probability Table (Portion)0.00= P(Z < 0.12)P(X < 8.6)

Upper Tail ProbabilitiesSuppose X is normal with mean 8.0 and standard deviation 5.0. Now Find P(X > 8.6)X8.68.0

Now Find P(X > 8.6)(continued)Z0.12 0Z0.120.5478 01.0001.0 - 0.5478 = 0.4522 P(X > 8.6) = P(Z > 0.12) = 1.0 - P(Z 0.12) = 1.0 - 0.5478 = 0.4522

Upper Tail Probabilities

Probability Between Two ValuesSuppose 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)Z0.12 0X8.6 8Calculate Z-values:

Z0.12Solution: Finding P(0 < Z < 0.12)0.04780.00= P(0 < Z < 0.12)P(8 < X < 8.6)= P(Z < 0.12) P(Z 0)= 0.5478 - .5000 = 0.04780.5000Z.00.010.0.5000.5040.5080.5398.54380.2.5793.5832.58710.3.6179.6217.6255.020.1.5478Standardized Normal Probability Table (Portion)

Suppose X is normal with mean 8.0 and standard deviation 5.0. Now Find P(7.4 < X < 8)X7.48.0Probabilities in the Lower Tail

Probabilities in the Lower Tail Now Find P(7.4 < X < 8)X7.48.0 P(7.4 < X < 8) = P(-0.12 < Z < 0)= P(Z < 0) P(Z -0.12)= 0.5000 - 0.4522 = 0.0478(continued)0.04780.4522Z-0.12 0The Normal distribution is symmetric, so this probability is the same as P(0 < Z < 0.12)

Steps to find the X value for a known probability:1. Find the Z value for the known probability2. Convert to X units using the formula:Finding the X value for a Known Probability

Finding the X value for a Known ProbabilityExample:Suppose X is normal with mean 8.0 and standard deviation 5.0. Now find the X value so that only 20% of all values are below this XX?8.00.2000Z? 0(continued)

Find the Z value for 20% in the Lower Tail20% area in the lower tail is consistent with a Z value of -0.84Z.03-0.9.1762.1736.2033-0.7.2327.2296.04-0.8.2005Standardized Normal Probability Table (Portion).05.1711.1977.2266X?8.00.2000Z-0.84 01. Find the Z value for the known probability

2. Convert to X units using the formula:Finding the X valueSo 20% of the values from a distribution with mean 8.0 and standard deviation 5.0 are less than 3.80

Evaluating NormalityNot all continuous random variables are normally distributedIt is important to evaluate how well the data set is approximated by a normal distribution

Evaluating NormalityConstruct charts or graphsFor small- or moderate-sized data sets, do stem-and-leaf display and box-and-whisker plot look symmetric?For large data sets, does the histogram or polygon appear bell-shaped?Compute descriptive summary measuresDo the mean, median and mode have similar values?Is the interquartile range approximately 1.33 ?Is the range approximately 6 ?(continued)

Assessing NormalityObserve the distribution of the data setDo 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 plotIs the normal probability plot approximately linear with positive slope?(continued)

The Normal Probability PlotNormal probability plotArrange data into ordered arrayFind corresponding standardized normal quantile valuesPlot the pairs of points with observed data values on the vertical axis and the standardized normal quantile values on the horizontal axisEvaluate the plot for evidence of linearity

A normal probability plot for data from a normal distribution will be approximately linear:306090-2-1012ZXThe Normal Probability Plot(continued)

Normal Probability PlotLeft-SkewedRight-SkewedRectangular306090-2-1012ZX(continued)306090-2-1012ZX306090-2-1012ZXNonlinear plots indicate a deviation from normality

The Uniform DistributionContinuous Probability DistributionsProbability DistributionsNormalUniformExponential

Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 6-*The Uniform DistributionThe uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable

Also called a rectangular distribution

Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.

The Continuous Uniform Distribution:wheref(X) = value of the density function at any X valuea = minimum value of Xb = maximum value of XThe Uniform Distribution(continued)f(X) =

Properties of the Uniform DistributionThe mean of a uniform distribution is

The standard deviation is

Uniform Distribution ExampleExample: Uniform probability distribution over the range 2 X 6:260.25f(X) = = 0.25 for 2 X 66 - 21Xf(X)

Uniform Distribution ExampleExample: Using the uniform probability distribution to find P(3 X 5):260.25P(3 X 5) = (Base)(Height) = (2)(0.25) = 0.5Xf(X)(continued)354

The Exponential DistributionContinuous Probability DistributionsProbability DistributionsNormalUniformExponential

The Exponential DistributionOften 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 dockTime between transactions at an ATM MachineTime between phone calls to the main operator

The Exponential DistributionDefined by a single parameter, its mean (lambda)The probability that an arrival time is less than some specified time X iswhere e = mathematical constant approximated by 2.71828 = the population mean number of arrivals per unitX = any value of the continuous variable where 0 < X <

Exponential Distribution ExampleExample: 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 = 15Three minutes is 0.05 hoursP(arrival time < .05) = 1 e-X = 1 e-(15)(0.05) = 0.5276So there is a 52.76% probability that the arrival time between successive customers is less than three minutes

Normal Approximation to the Binomial DistributionThe binomial distribution is a discrete distribution, but the normal is continuousTo use the normal to approximate the binomial, accuracy is improved if you use a correction for continuity adjustmentExample: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 DistributionThe closer p is to 0.5, the better the normal approximation to the binomialThe 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

np 5 andn(1 p) 5 (continued)

Normal Approximation to the Binomial DistributionThe mean and standard deviation of the binomial distribution are = np

Transform binomial to normal using the formula: (continued)

Using the Normal Approximation to the Binomial DistributionIf n = 1000 and p = 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) = 0.0618X180.5200-1.540Z

Chapter SummaryPresented key continuous distributions normal, uniform, exponentialFound probabilities using formulas and tablesRecognized when to apply different distributions Applied distributions to decision problems