ch 6 the normal distribution

The Normal Distribution

Chapter 6

Objectives

• By the end of this presentation, you should be able to:1. Recognize the normal probability distribution and apply it

appropriately

2. Recognize the standard normal probability distribution and apply it appropriately

3. Compare normal probability by converting to the standard normal distribution

Normal Distribution Standard Normal Distribution Using the Normal Distribution


• The normal distribution is a continuous distribution

• The graph of the normal distribution is bell shaped

• It is unimodal and symmetric



• The mean, median, and mode of the normal distribution are all equal and located at the peak

• The normal distribution is symmetrical about its mean

• Half the area under the curve is above the peak, and the other half is below it



• Let X be a normal random variable

• X~N(Ã,Ç)• The normal probability density

function is

• the population mean• the population standard deviation• 3.141593…• 2.718281…

• The probability that is

• We will learn to use a z-table to calculate the probabilities



• IMPORTANT: The normal distribution depends on knowing the mean , and the standard deviation

• The value of the mean, , tells you where the normal distribution will be centered

• The value of the standard deviation tells you how spread out (or wide) the distribution will be

• Large standard deviations mean the range of the normal distribution will be bigger, and the peak will be lower

• Small standard deviations mean the range of the normal distribution will be small, and the peak will be higher


Example 1

• The normal distributions shown have means of , , and ; Identify which graph has which mean

• The normal distributions shown have standard deviations of , 1, and 2; identify which graph has which standard deviation


The Standard Normal Distribution• A normal distribution with a mean of 0

and a standard deviation of 1 is called the standard normal distribution

• So a standard normal random variable is Z ~ N(0, 1)

The Standard Normal Distribution

• Recall: The formula for the number of standard deviations away from the mean

• This actually has a name: z score

• The z score tells you how many standard deviations the value x is above (to the right) or below (to the left) of the mean


The Standard Normal Distribution

• The standard normal distribution corresponds to z scores

• Positive z scores tell us the value of x was larger than the mean

• Negative z scores tell us the value of x was smaller than the mean


Areas under the Normal Distribution – Empirical Rule• About 68 percent of the area under the normal curve is within one standard deviation

of the mean

• About 95 percent is within two standard deviations of the mean

• 99.7 percent is within three standard deviations of the mean


Areas under the Normal Distribution – Empirical Rule

• About 68 percent of the area under the normal curve is within one standard deviation of the mean



• About 95 percent is within two standard deviations of the mean



• 99.7 percent is within three standard deviations of the mean


Example 2

• The daily water usage per person in a town is normally distributed with a mean of 20 gallons and a standard deviation of 5 gallons

• Question: About 68% of the daily water usage per person lies between what two values?

• 68% corresponds to , which we will write: • Answer: • That is, about 68% of the daily water usage will lie between 15 and

25 gallons


Example 2

• The daily water usage per person in a town is normally distributed with a mean of 20 gallons and a standard deviation of 5 gallons

• Question: What percentage of the daily water usage per person lies between 10 and 30 gallons?

; ; • So, 10 and 30 gallons are within 2 standard deviations each

way, this is what percent of the data?• 95%


Using the Normal Distribution• The shaded area indicates the area

to the left of x• This area is a probability• The shaded area represents the

probability • The area to the right is then

• How do we calculate these probabilities? z-tables!



to the right of x• This area is a probability• The shaded area represents the

probability • The area to the right is




between a and b• This area is a probability• The shaded area represents the

probability • The area to the right is then



Using the Normal Distribution

• The area to the right is then

Why?


Calculating Probabilities from a Normal Distribution• Here is the general procedure to calculate probabilities

from a normal distribution1. You are given an interval in terms of x

2. Convert to a z-score by using

3. Look up probability in z-table that corresponds to z score

• How does a z-table work?


Example 3

• The amount of tip the servers in an exclusive restaurant receive per shift is normally distributed with a mean of $80 and a standard deviation of $10. Suzy feels she has provided poor service if her tip for the shift is less than $65. What percentage of the time will she feel like she provided poor service?

• Find


Example 3


• Converting to z-score• Now, look up on the z-table = .0668

• Suzy will feel like she has provided poor service 6.68% of the time.


Example 3 (with technology)


• Find • 2nd DISTR (above VARS)• 2:normalcdf(

• Where is the min?• How do we say this?


Example 4

• The weights of melons harvested at a farm are normally distributed with a mean 5 pounds and a standard deviation of 0.5 pounds. Find the probability that a randomly selected melon weighs between 4 and 6 pounds.


Example 4


• Converting to z-score• For x = 4

• For x = 6•


Example 4


• Now, we’ll look up

• Next, we’ll look up

• =.9772-.0228• =.9544• just like the Empirical Rule!




• With Technology• 2nd DISTR (above VARS)• 2:normalcdf(• normalcdf(4, 6, 5, 0.5)


Percentile Calculations Based on the Normal Distribution• Here is the general procedure to calculate the value x that

corresponds to the Pth percentile.1. You are given a probability or percentile desired2. Look up the z-score in the table that corresponds to the

probability3. Convert to x by the following formula

(where did this come from?)


Example 5

• The daily water usage per person in a town is normally distributed with a mean of 20 gallons and a standard deviation of 5 gallons. A special tax is going to be charged on the top 5% of water users. Find the value of daily water usage that generates a special tax.

• (What z value corresponds to 95%?)

x

95%

5%

A special tax will be charged on customers who use more than 28.85 gallons per day



• The daily water usage per person in a town is normally distributed with a mean of 20 gallons and a standard deviation of 5 gallons. A special tax is going to be charged on the top 5% of water users. Find the value of daily water usage that generates a special tax.

• 2nd DISTR; 3:invNorm(• =28.22% (why different?)

x

95%

5%

A special tax will be charged on customers who use more than 28.85 gallons per day


Homework

• Page 363: 60-64, 70-72, 76-78, 86


ch 6 the normal distribution

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