Slide 1:STA/MTH 379 Dr. Ananda Bandulasiri Assistant Professor of Statistics SHSU

Slide 2:Chapter 7 Population Distributions

Slide 3:Definitions A variable associates a value with each individual or object in a population. A variable can be either categorical or numerical. The distribution of all values of a numerical variable or all the categories of a categorical variable including probabilities is called a population distribution.

Slide 4:Numerical Variables & Probability A discrete numerical variable is one whose possible values are isolated points along the number line. A continuous numerical variable is one whose possible values form an interval along the number line. The population distribution for a discrete numerical variable can be summarized by a relative frequency histogram. The population distribution for a continuous numerical variable is summarized by a density histogram.

Slide 5:Example

Slide 6:Example - continued

Slide 7:Probability Histogram

Slide 8:Continuous Probability Distributions

Slide 9:Continuous Probability Distributions We must use density histograms (or density curves) when describing continuous variables. This preserves proportions. The probability that x takes a value in any interval is the appropriate area under the density histogram (or density curve).

Slide 10:Some Illustrations

Slide 11:Illustrations

Slide 12:Population Parameters The mean value of a numerical variable x, denoted by , describes where the population distribution of x is centered. The standard deviation of a numerical variable x, denoted by , describes variability in the population distribution. When it is close to zero, the values of x tend to be close to the mean. When it is large, there is more variability in the population of x values.

Slide 13:Illustrations Two distributions with the same standard deviation with different means.

Slide 14:Illustrations Two distributions with the same mean and different standard deviations.

Slide 15:Example with Density Distributions Suppose that we are interested in the weight of Priority Mail Packages so that x = package weight (in pounds) After careful study, we decide that the following continuous probability distribution provides a good model for the package weights.

Slide 17:The probability that a randomly selected package weighs more than 1 pound is the area under the curve to the right of 1.

Slide 18: The Normal Distribution Normal distributions are continuous probability distributions that are bell shaped and symmetric. Normal distributions are referred to as normal curves. Normal curves provide reasonable approximations to the distributions of many different variables. Normal curves play a central role in many procedures in inferential statistics.

Slide 19:Normal Distributions Two characteristic values (parameters) completely determine a normal distribution Population Mean = µ Population Standard Deviation = ?

Slide 24:Major Principle The proportion or percentage of a normally distributed population that is in an interval depends only on how many standard deviations the endpoints are from the mean.

Slide 25:Standard Normal Distribution A normal distribution with mean = 0 and standard deviation = 1, is called the Standard Normal (Z) distribution.

Slide 26:Standard NormalTables

Slide 27:Standard NormalTables

Slide 28:Using the Normal Tables Find P( z < 0.46)

Slide 29:Using the Normal Tables Find P(z < -2.74)

Slide 30:Calculations Using the Standard Normal Distribution

Slide 31:Sample Calculations

Slide 32:Symmetry Property From the preceding examples that it becomes obvious that P( z > z*) = P( z < -z*)

Slide 33:Sample Calculations Using the Standard Normal Distribution

Slide 34:Sample Calculations Using the Standard Normal Distribution

Slide 35:Sample Calculations Using the Standard Normal Distribution

Slide 36:Example Calculation

Slide 37:Example Calculation

Slide 38:Finding Normal Probabilities To calculate probabilities for ANY normal distribution, standardize the relevant values and then use the standard normal table. More specifically, if x is a variable whose behavior is described by a normal distribution with mean m and standard deviation s, (i.e. x ~ N(m , s ), then P(x < b) = p(z < b*) P(x > a) = P(z > a*) where z is a variable whose distribution is standard normal and

Slide 39:Standard Normal Distribution Revisited

Slide 40:Conversion to N(0,1)

Slide 41:Example 1

Slide 42:Example 1 What proportion of the jars are under-filled (i.e., have less than 20 ounces of sauce)?

Slide 43:Example 1

Slide 44:Example 1

Slide 45:Example 1

Slide 46:The weight of the cereal in a box is a normal variable with mean 12.15 ounces, and standard deviation 0.2 ounce. What percentage of the boxes have contents that weigh under 12 ounces? Example 2

Slide 47:If the manufacturer claims there are 12 ounces in a box, does this percentage cause concern? If so, what could be done to correct the situation? The machinery could be reset with a higher or larger mean. The machinery could be replaced with machinery that has a smaller standard deviation of fills. The label on the box could be changed. Which would probably be cheaper immediately but might cost more in the long run? In the long run, it might be cheaper to get newer more precise equipment and not give away as much excess cereal. Example 2

Slide 48:Homework

Slide 49:Normal Plots

Slide 50:Normal Probability Plot Example

Slide 51:Normal Probability Plot Example

Slide 52:Normal Probability Plot Example

Slide 53:Normal Probability Plot Example