Ch. 6 The Normal Distribution 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 measure accurately.

The polygon of the distribution become a smooth curve called Probability Density Function (this is equivalent to Probability Distribution for discrete random variable) f(X) P ( a ≤ X ≤ b ) = P ( a < X < b ) a b X (Note that the probability of any individual value is zero)

The Normal Distribution Bell Shaped Perfectly Symmetrical Mean, Median and Mode are Equal Location on the value axis is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: +  to   f(X) σ +  μ   Mean = Median = Mode

What can we say about the distribution of values around the mean What can we say about the distribution of values around the mean? There are some general rules: Probability is measured by the area under the curve The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below The height of the curve at a certain value below the mean is equal to the height of the curve at the same value above the mean The area between: μ ± 1σ encloses about 68% of X’s μ ± 2σ covers about 95% of X’s μ ± 3σ covers about 99.7% of X’s

The Normal Distribution Density Function: The formula for the normal probability density function is: (the curve is generated by): 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 Note: f (X) is the same concept as P(X).

By varying the parameters μ and σ, we obtain different normal distributions. That is to say that any particular combination of μ and σ, will produce a different normal distribution. This means that we have to deal with numerous distribution tables. Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread.

Standard Normal Distribution Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardized normal distribution (Z). To do so, we need to transform all X units (values) into Z units (values) Then the resulting standard normal distribution has a mean=zero and a standard deviation =1 X-Values above the mean of X will have positive Z-value and X-values below the mean of will have negative Z-values The transformation equation is:

EXAMPLE: A sample of 19 apartment’s electricity bill in December is distributed normally with mean of $65 and std. dev. Of $9. Z- values for X = $83 is: Z= (83-65)/9=18/9=2. That is to say that X=83 is 2 standard deviation above the mean or $65. Note that the distributions are the same, only the scale has changed. X (μ = 65, σ =9) 47 56 65 74 83 Z (μ = 0, σ =1) -2 -1 1 2

Applications: Use of Standard Normal Distribution Table, Pages 737-738 What is the probability that a randomly selected bill is between $56 and $74?

What is the value of the lower 10% of the bills What is the value of the lower 10% of the bills? (finding X value for a known probability)

What is the value of the highest 10% of the bills What is the value of the highest 10% of the bills? (finding X value for a known probability)

What is the range of the values that contain the middle 95% of the bills?

Assessing Normality Not all continuous random variables are normally distributed It is important to evaluate how well the data set is approximated by a normal distribution Construct charts or graphs For 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 measures Do the mean, median and mode have similar values? Is the interquartile range approximately 1.33 σ? Is the range approximately 6 σ? Observe the distribution of the data set within certain std. Dev. Evaluate normal probability plot Is the normal probability plot approximately linear with positive slope? Page 209