1. Chapter 5: Normal Probability Distributions
Lesson 5.1: Introduction to Normal Distributions and the Standard Normal Distribution

2. Normal Distribution
A normal distribution is a continuous probability distribution for a random variable x.
Properties
Mean, median, and mode are equal
Bell-shaped and symmetric about the mean
Total area under the curve is equal to 1
As the curve extends farther from the mean, it gets closer to the x-axis but never touches it. (x-axis is an asymptote)
The points at which the curvature changes are called inflection points (one standard deviation from the mean)

3. Parameters
The shape of the normal curve depends on the mean (center) and standard deviation (spread)
Same Mean
Different Standard Deviation
Different Mean
Same Standard Deviation

4. Mean and Standard Deviation

5. Empirical Rule
About 68% of the data lies within 1 standard deviation of the mean About 95% of the data lies within 2 standard deviations of the mean About 99.7% of the data lies within 3 standard deviations of the mean

6. Empirical Rule Example
Each portion of the SAT is designed to be approximately normal and have an overall mean of 500 and standard deviation of 100.
What percent of students will score above 700?
What percent of students will score below 400?
What percent of students will score between 600 and 800? .

7. The Standard Normal Distribution
The Standard Normal Distribution has
a mean of 0
a standard deviation of 1
Probabilities can be found from the standard normal distribution table in the back of your book.

8. Finding Probabilities less than Z
P(z < 0.78) =
P(z < 1.4) =
P(z < 0.21) =

9. Finding Probabilities more than Z
P(z > 1.23) = 1 – =
P(z > 0.07) =
P(z > 0.95) =

10. Finding Probabilities between Z scores
P(0.41 < z < 0.93) = – =
P(0.64 z < 1.07) =
P(0.88 < z < 0.72) =

11. Chapter 5: Normal Probability Distributions
Lesson 5.2: Normal Distributions: Finding Probabilities

12. Finding Probabilities with the table
Female heights are normally distributed with a mean of 65 inches and a standard deviation of 2 inches
What is the probability that you randomly select a female who is shorter than 66.5 inches?
Calculate a z-score:
Z = (66.5 – 65)/2 = 0.75
P(x < 66.5) = P(z < 0.75) =

13. Finding Probabilities with technology
[2nd ][vars][normalcdf]
normalcdf (xL,xU,μ,σ)
This function returns the probability of a normal distribution from a lower value (xL) to an upper value (xU) given a mean (μ) and a standard deviation (σ)
Example: Assume that cholesterol levels of men in the US are normally distributed with a mean of 215 and standard deviation of 25 milligrams per deciliter. If you randomly select a man from the US, what is the probability that his cholesterol level is
less than 175
more than 260
between 200 and 240

14. Chapter 5: Normal Probability Distributions
Lesson 5.3: Normal Distributions: Finding Values

15. Finding z-scores Find the z-score that …
has a cumulative area of
has 10.75% of the distribution’s area to the right
corresponds to the first quartile
corresponds to P17

16. Finding Values
Example: Monthly utility bills are normally distributed with a mean of $100 and a standard deviation of $12.
What is the smallest utility bill that can be in the top 10% of the bills?
What is the value of the utility bill that represents the first quartile?
What is the largest utility bill that can be in the bottom 35% of the bills?
What are the cut off values for the middle 70% bills?

17. Chapter 5: Normal Probability Distributions
Lesson 5.4: Sampling Distributions and the Central Limit Theorem

18. Sampling Distribution for a Mean
A sampling distribution is the probability distribution of a sample statistic that is formed when samples of size n are repeatedly taken from a population
Properties
μx = μ
σx = σ/√n

19. The Central Limit Theorem
If samples of size 30 or bigger are taken from any population then the sampling distribution of sample means approximates a normal distribution.
If the population itself is normally distributed, then the sampling distribution of sample means is normally distributed for any sample size.

20. Finding Probabilities for x and x
Monthly rent for a two bedroom apartment is normally distributed with a mean of $850 and standard deviation of $75. What is the probability that…
a randomly selected apartment costs more than $890 a month?
30 randomly selected apartments on average costs more than $890 a month?
Credit card debit for college students is normally distributed with a mean of $4100 and a standard deviation of $800.
Would it be unusual to randomly select a college student with $3000 in credit card debt?
Would it be unusual to randomly select twenty college students with $3000 in credit card debt?