Chapter 7 Lesson 7.6 Random Variables and Probability Distributions 7.6: Normal Distributions

Chapter 7Lesson 7.6

Random Variables and Probability Distributions7.6: Normal Distributions

Special Distributions

Two Discrete Distributions:Binomial and Geometric

One Continuous Distribution:Normal Distributions

Normal Distributions• Continuous probability distribution• Symmetrical bell-shaped (unimodal) density curve

defined by the mean and standard deviation.• Area under the curve equals 1• Probability of observing a value in a particular

interval is calculated by finding the area under the curve

• Normal models are appropriate for distributions whose shapes are unimodal and roughly symmetric.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Normal Model

There is a Normal model for every possible combination of mean and standard deviation. We write N(μ,σ) to represent a Normal model

with a mean of μ and a standard deviation of σ.

Standard Normal Distribution

• Is a normal distribution with Mean = 0 & SD = 1• N(0,1)• We can standardized any normal

distribution by using z-scores

The 68-95-99.7 Rule (cont.)

• The following shows what the 68-95-99.7 Rule tells us:


Finding Normal Percentiles by Hand

When a data value doesn’t fall exactly 1, 2, or 3 standard deviations from the mean, we can look it up in a table of Normal percentiles.

Appendix A Table 2 (pg.862) provides us with normal percentiles, but many calculators and statistics computer packages provide these as well.


Finding Normal Percentiles by Hand (cont.)

Table A is the standard Normal table. *We have to convert our data to z-scores before using the table.*

This is how to find the area to the left when we have a

z-score = 1.80


Finding Normal Percentiles Using Technology (cont.)

Many calculators and statistics programs have the ability to find normal percentiles for us.

Take out the TI!Take out the TI!

Suppose we are interested in the probability that z* is greater than 2.31.

P(z > 2.31) =

1 - .9896 = .0104


From Percentiles to Scores: z in Reverse

Reverse! Sometimes we start with areas and need to find the corresponding z-score.

Example: What z-score represents the first quartile in a Normal model?


From Percentiles to Scores: z in Reverse (cont.)

Take out the TI!Take out the TI!

Suppose we are interested in the finding the z* for the smallest 2%.

P(z < z*) = .02

z* = -2.054z*

Suppose we are interested in the finding the z* for the largest 5%.

P(z > z*) = .05

z* = 1.645z*

.95

Data on the length of time to complete registration for classes using an on-line registration system suggest that the distribution of the variable

x = time to register

for students at a particular university can well be approximated by a normal distribution with mean m = 12 minutes and standard deviation s = 2 minutes.

Registration Problem Continued . . .


m = 12 minutes and s = 2 minutes

What is the probability that it will take a randomly selected student less than 9 minutes to complete registration?

P(x < 9) =

Normalcdf(-99,-1.5)

Standardized this value.

.0668

9




What is the probability that it will take a randomly selected student more than 13 minutes to complete registration?

P(x > 13) =

Normalcdf(0.5, 99)

Standardized this value.

.3085

13




What is the probability that it will take a randomly selected student between 7 and 15 minutes to complete registration?

P(7 < x < 15) =

Standardized these values.

.9270

7 15




Because some students do not log off properly, the university would like to log off students automatically after some time has elapsed. It is decided to select this time so that only 1% of students will be automatically logged off while still trying to register.

What time should the automatic log off be set at?

P(x > z*) = .01

z* = 16.66

z*

.01

212

33.2

x

.99

Homework

• Pg.457: #7.63, 65, 67, 68, 70, 72abcd, 75, 81

Documents

Chapter 7 Lesson 7.6 Random Variables and Probability Distributions 7.6: Normal Distributions