Chapter 6 The Normal Distribution

Density Curves

bell shaped curve for a dist of data

Properties of Density Curves (2)

• is always on or above the horizontal axis

• the total area under a density curve (and above the horizontal axis) equals 1

Normally Distributed Variables

a variable is said to have norm dist if it has the shape of a normal curve

Population norm dist vs Approximately norm dist

• if population is said to be norm dist its a norm dist population

• if a population is said to be a approx norm dist then it has a approx norm dist

Normal Dist Characteristics

• bell shaped

• centered at mean

• the norm curve is close to the horizontal axis outside the range of mean-3SD to mean+3SD

Standard Norm Distribution and Standard Normal Curve

• has mean 0

• has SD 1

Standardized Normally Distributed Variable

z=x-mean/SD

To Find Percentages for a Norm Distributed Variable

1) expressing the range in terms of z-scores

2) determining the corresponding area under the standard normal curve

Properties of the Standard Normal Curve (4)

• the total area under the standard normal curve is 1

• the standard normal curve extends indefinitely in both directions, approaching but never touching the horizontal axis as it does so

• symmetric around 0

• almost all the area under the curve lies between -3,3

Using the standard norm table

• areas under the standard normal curve are so important that we have tables of those areas

• you are given z score then use z-score to find area at that point

• left page is neg z-score

• right page is positive z-score

Za notation

• Za is used to show that z score has a area of a to the right under the standard norm curve (Za= Z sub a)

To determine a percentage of probability for a norm dist variable (4)

• sketch norm curve associated with variable

• shade region and mark x values

• find z scores for those x values

• use table to find area under standard normal curve by z scores found previously

Empirical Rule for Variables (3)

Property 1: Approximately 68% of all possible observations lie within one SD to either side of the mean

Property 2: Approximately 95% of all possible observations lie within two SD to either side of the mean

Property 3: Approximately 90.7% of all possible observations lie within three SD to either side of the mean

Determining the observations corresponding to s specified percentage or prob for a norm dist variable

• sketch the norm curve associated with the variable

• shade region of interest

• use table to determine z score

• find x values using z scores found previously USE formular: x=mean+z(SD)

• the form is just z score rewritten

Assessing Normality

• normal probability plot, plots the observed values and z-scores which are the observations expected for the variable to have a normal distribution

Guidelines for Assessing Normality Using a Norm Prob plot

• if plot is roughly linear you can assume that the variable is approximately normally distributed

• if the plot is not roughly linear, you can assume that the variable is not approximately norm distributed

Normal Probability Plot

a scatter plot with the ranked data values on one axis and their corresponding expected z-scores from a standard norm dist on the other axis

How to approximate binomial probabilities by Normal Curve Areas

• Find n, the number of trials, and p, the success prob

• continue only if both np and n(1-p) are 5 or greater

• find mean and SD using the binomial dist formulas

• make correction for continuity and find required area under curve

Correction for Continuity

• adjust boundaries by +- 0.5 when switching from discrete (binomial) to continuous (normal)

• Replace any whole number x with the interval

  x−0.5 to x+0.5