Ch 6: The Normal Distribution and Other Continuous Distributions

Mean of Uniform Distribution

(a + b) / 2

Standardized Normal Distribution

Also known as "Z" distribution Mean is 0 Standard deviation is 1 Values above mean have positive Z values Values below mean have negative Z values

Standardized Normal

Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardized normal distribution (Z) To compute normal probabilities need to transform X units into Z units Standardized normal distribution (Z) as a mean of 0 and a standard deviation of 1

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

Normal Distribution

Bell shaped Symmetrical Mean, median, and mode are equal Location is determined by the mean Spread is determined by the standard deviation Random variable has an infinite theoretical range

Normal Distribution Shape

Changing the mean shifts the distribution left or right Changing the standard deviation increases or decreases the spread

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

Steps to Find the X Value for a Known Probability

Find Z value for the known probability Convert to X units using the formula X= mean + Z*standard deviation

Standardized Normal Table

Gives the probability less than a desired value of Z (ex: from negative infinity to Z) Row- shows value of Z to the first decimal point Column- gives value of Z to the second decimal point Value within the table gives the probability from Z= - infinity up to the desired Z value

Evaluating Normality

Not all continuous distributions are normal Important to evaluate how well the data set is approximated by a normal distribution Normally distributed data should approximate the theoretical normal distribution

Exponential Distribution

Often used to model the length of time between 2 occurrences of an event (the time between arrivals) Time between trucks arriving at an unloading dock

Exponential Probability Density Formula

P(arrival time < X) = 1 - e^-lambda * X

Uniform Distribution

Probability distribution that has equal probabilities for all possible outcomes of the random variable Rectangular distribution

Probability as Area Under the Curve

Total area under the curve= 1 Symmetric so half is above mean, half below

Translation into 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

Continuous Random Variable

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 These can take on any value depending only on the ability to precisely and accurately measure

Continuous Uniform Distribution Formula

f(x) = { [ 1 / (b-a) if a < or equal to X < or equal to b 0 otherwise

Empirical Rule

mean +/- 1 standard deviation encloses about 68% of X's mean +/- 2 standard deviations encloses about 95% of X's mean +/- 3 standard deviations encloses about 99% of X's

Normal Distribution Density Function

see slides

Standardized Normal Probability Density Function

see slides

Standard Deviation of a Uniform Distribution

sqrt [ (b - a)^2 / 12 ]