Chapter 6

ContinuousDistributions

Understanding concepts of the continuous distribution, especially the normal distribution.Recognizing normal distribution problems, and knowing how to solve them.Deciding when to use the normal distribution to approximate binomial distribution problems, and how to work them.Decidong when to use the exponential distribution to solve problems in business, and know how to work them.*Learnings

Continuous DistributionsContinuous distributionsContinuous distributions are constructed from continuous random variables which can be any values over a given intervalWith continuous distributions, probabilities of outcomes occurring between particular points are determined by calculating the area under the curve between these pointsUnlike discrete probability distributions, the probability of being exactly at a given point is 0 (since you can measure it more precisely)*

Properties of the Normal DistributionCharacteristics of the normal distribution:Continuous distribution - Line does not breakBell-shaped, symmetrical distribution

Ranges from - to Mean = median = modeArea under the curve = total probability = 168% of data are within one std dev of mean, 95% within two std devs, and 99.7% within three std devs*

Probability Density Function ofthe Normal DistributionThere are a number of different normal distributions, they are characterized by the mean and the std dev*

Probability Density Function ofthe Normal Distribution*

Rather than create a different table for every normal distribution (with different mean and std devs), we can calculate a standardized normal distribution, using the transformation of the normal random variable x into standard normal variate, Z

A z-score gives the number of standard deviations that a value x is above the mean.Z distribution is normal distribution with a mean of 0 and a std dev of 1 Normal Distribution Calculating Probabilities*

Standardized Normal Distribution - ContinuedZ distribution probability values are given in table A5 or can be calculated using softwareTable A5 gives the total area under the Z curve between 0 and any point on the positive Z axisSince the curve is symmetric, the area under the curve between Z and 0 is the same whether the Z value is positive or negative*

Z TableSecond Decimal Place in Z Z0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.000.00000.00400.00800.01200.01600.01990.02390.02790.03190.03590.100.03980.04380.04780.05170.05570.05960.06360.06750.07140.07530.200.07930.08320.08710.09100.09480.09870.10260.10640.11030.11410.300.11790.12170.12550.12930.13310.13680.14060.14430.14800.1517

0.900.31590.31860.32120.32380.32640.32890.33150.33400.33650.33891.000.34130.34380.34610.34850.35080.35310.35540.35770.35990.36211.100.36430.36650.36860.37080.37290.37490.37700.37900.38100.38301.200.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015

2.000.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817

3.000.49870.49870.49870.49880.49880.49890.49890.49890.49900.49903.400.49970.49970.49970.49970.49970.49970.49970.49970.49970.49983.500.49980.49980.49980.49980.49980.49980.49980.49980.49980.4998

*

Table Lookup of a StandardNormal Probability Z0.00 0.01 0.02

0.000.00000.00400.00800.100.03980.04380.04780.200.07930.08320.0871

1.000.34130.34380.34611.100.36430.36650.36861.200.38490.38690.3888*

Applying the Z Formula*

Applying the Z Formula0.5 + 0.2123 = 0.7123*

Applying the Z Formula0.5 0.4803 = 0.0197*

Applying the Z Formula*

Demonstration Problem 6.9These types of problems can be solved quite easily with the appropriate technology. The output shows the MINITAB solution. Suppose we know that X is normally distributed with mean 3.58 and std dev 1.04, and we want P(X

Normal Approximation of theBinomial DistributionFor certain types of binomial distributions, the normal distribution can be used to approximate the probabilitiesAt large sample sizes, binomial distributions approach the normal distribution in shape regardless of the value of pThe normal distribution is a good approximate for binomial distribution problems for large values of n*

Normal Approximation of Binomial:Parameter ConversionConversion equations

Conversion example:*

Normal Approximation of Binomial:Interval Check*

Normal Approximation of Binomial:Correcting for Continuity *

Normal Approximation of Binomial: Computations*

Another common continuous distribution is the exponential distributionExample the time until a light bulb works is known to follow an exponential distributionIts skewed to the rightApex is always at X = 0Steadily decreases as X gets largerExponential Distribution*

Different Exponential Distributions*

Exponential Distribution: Probability Computation*

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