9/19/2016

Mathematical Models for Data

A mathematical model for data, commonly referred to as a density curve or p.d.f., provides a way to describe the entire distribution with a single expression.

A density curve is a function f(x) that satisfies:

1. f(x)>=0 for all x
2. The total area under the curve f(x) is one.

Idealized center and spread

• The density curve is an idealized description of the distribution of the data.
• The mean, or balancing point, of the density curve is denoted by the Greek letter mu.
• The median of the density curve is the value that has area .5 to the right and .5 to the left.
• The standard deviation of the density curve measures the variability from the idealized mean, and is denoted by the Greek letter sigma.

Normal Distributions

• Normal Distributions are symmetric, single-peaked, bell-shaped density curves.
• All normal distributions have the same overall shape.
• The exact density curve for a particular normal distribution is given by specifying mu and sigma. 

The Empirical Rule  In any normal distribution:

1. 68% of the observations fall within 1 standard deviation of the mean.
2. 95% of the observations fall within 2 standard deviations of the mean.
3. 99.7% of the observations fall within 3 standard deviations of the mean.

Calculations for Normal Distributions

• If X~N(mu, sigma) then the standardized variable Z=(X-mu)/sigma~N(0,1).
• You may always use RStudio to do calculations for normal distributions. The commands pnorm(x, mean, sd), xpnorm(x, mean, sd), qnorm(x, mean, sd), and xqnorm(x, mean, sd) will be very helpful.

Examples

• Find the area under the standard normal curve to the left of -1.4.
• Find the area under the N(0,1) curve between .76 and 1.4.
• Find the value, z, of the N(0,1) distribution which has area .25 to the right of z.
• Suppose X~N(275, 43).  Find P(X>200).
• Suppose X~N(275, 43).  Find P(200<X<375).
• Suppose verbal SAT scores follow the N(430, 100) distribution.  How high must a student score in order to place in the top 5%?

Please read of Sections 3.2 and 3.3 for class on Wednesday.

Reminder: Our first quiz covering Chapters 1 and 2 will be in class on Friday.