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MATH 170 - Applied Statistics Section 7.6 Normal Distribution – Part I
1 The Normal Distribution
We’ve dealt with normal curves before ... recall what they look like.
Definition 1 (Normally Distributed Random Variable) A continuous random variable x is said to have a normal distribution if its density curve is a normal curve. Features of the density curve for a normally distributed random variable are:
- The mean value, μ, determines where the curve is centered
- The standard deviation value, σ, determines the extent to which the curve spreads out about μ (there is a change in concavity of the density curve at μ − σ and μ + σ.
2 The Standard Normal Distribution
An infinite number of combinations of μ and σ exist, and we concentrate on a very simple μ, σ pairing.
Definition 2 (Standard Normal Distribution) A continuous random variable, z, is said to have a standard normal distribution if it has a normal distribution with mean μ = 0 and standard deviation σ = 1. The corresponding density curve is referred to as the standard normal or z-curve.
The z-Curve
3 Using a Table to Find Areas Under the Standard Normal Curve
We’ve mentioned before that the area under a normal curve is not one we can calculate using simple geometry formulas. So we’ll use two different methods for determining the area under a portion of the standard normal curve: first, using a look-up-table, and second, using the TI-83.
Let c denote a number between -3.89 and 3.89 (as high and low as the table goes) that has two digits to the right of the decimal point. For such a c, the table in the book’s back page gives the area to the LEFT of c under the normal curve, i.e.
area under the curve and to the LEFT of c m cumulative area to the left of c m P (z < c) = P (z ≤ c)
P (z < c) = P (z ≤ c)
To read the probability from the table, locate:
- The row labeled with the sign of c and the digits to either side of the decimal point. (Ex: 2.3 for c = 2.37 or -0.6 for c = − 0 .69)
- The column identifying the second digit to the right of the decimal point in c. (Ex: If c = 2.37, look down the column .07)
The desired probability (area) is the number at the intersection of this row and column. The illustration below shows how to look up P (z ≤ 2 .37) from the table:
P (z ≤ 2 .37) = 0. 9911
4 Using the TI-83 to Find Areas Under the Standard Normal Curve
Rather using a look-up table, we can use the TI-83’s Distribution Functions (DISTR)to help us determine probabilities associated with a normal variable. The key idea you want to remember is:
P (a ≤ x ≤ b) = normalcdf( lower bound, upper bound, μ, σ )
In other words, normalcdf (with a CDF, not the one with the PDF) computes the probability a normally distributed random variable x is between two different values.
The normalcdf function is found under the DISTR menu (2nd^ VARS) and is option number 2.
Given the fact that we must express our probability as being between TWO values when using normalcdf, here’s how we handle other situations. Assume x is a normally distributed random variable with mean μ and standard deviation σ, then:
- Probability x is less than a value:
P (x < b) = P (x ≤ b) ≈ P ( really big negative number < x < b)
= normalcdf(really big negative number, b, μ, σ)
- Probability x is greater than a value: P (x > a) = P (a < x) ≈ P (a < x < really big positive number )
= normalcdf(a, really big positive number, μ, σ)
Example 2 Use the normalcdf function on your calculator to determine the following probabilities associated with the standard normal variable z. Indicate what you entered on your calculator to get this value!
- P (z ≤ − 0 .67)
- P (− 1. 23 ≤ z < 0 .85)
- P (0 < z < 2)
- P (z ≥ 1 .59)
Homework: pp. 407 - 409: # 65, 67
