1. BCOR 1020 Business Statistics
Lecture 12 – February 26, 2008

2. Overview Chapter 7 – Continuous Distributions Continuous Variables
Describing a Continuous Distribution
Uniform Continuous Distribution
Normal Distribution
Standard Normal Distribution

3. Chapter 7 – Continuous Variables
Events as Intervals:
Discrete Variable – each value of X has its own probability P(X).
Continuous Variable – events are intervals and probabilities are areas underneath smooth curves. A single point has no probability.

4. Chapter 7 – Continuous Variables
PDFs and CDFs:
Probability Density Function (PDF) – For a continuous random variable, the PDF is an equation that shows the height of the curve f(x) at each possible value of X over the range of X.
Normal PDF

5. Chapter 7 – Continuous Variables
PDFs and CDFs:
Continuous PDF’s:
Denoted f(x)
Must be nonnegative
Total area under curve = 1
Mean, variance and shape depend on the PDF parameters
Reveals the shape of the distribution
Normal PDF

6. Chapter 7 – Continuous Variables
PDFs and CDFs:
Continuous Cumulative Distribution Functions (CDF’s):
Denoted F(x)
Shows P(X < x), the cumulative proportion of scores
Useful for finding probabilities
Normal CDF

7. Chapter 7 – Continuous Variables
Probabilities as Areas:
Continuous probability functions are smooth curves.
Unlike discrete distributions, the area at any single point = 0.
The entire area under any PDF must be 1.
Mean is the balance point of the distribution.

8. Chapter 7 – Continuous Variables
Expected Value and Variance:

9. Chapter 7 – Normal Distribution
Characteristics of the Normal Distribution:
Normal or Gaussian distribution was named for German mathematician Karl Gauss (1777 – 1855).
Denoted N(m, s)
“Bell-shaped” Distribution
Domain is – < X < + 
Defined by two parameters, m and s
Symmetric about x = m
Almost all area under the normal curve is included in the range m – 3s < X < m + 3s (Recall the Empirical rule.)

10. Chapter 7 – Normal Distribution
When does a random variable have a
Normal distribution?
It is assumed in our experiment or problem.
Our variable is the sample average for a large sample. (We will discuss why later.)
A normal random variable should:
Be measured on a continuous scale.
Possess clear central tendency.
Have only one peak (unimodal).
Exhibit tapering tails.
Be symmetric about the mean (equal tails).

11. Chapter 7 – Normal Distribution
Characteristics of the Normal Distribution:

12. Chapter 7 – Normal Distribution
Characteristics of the Normal Distribution:
Normal PDF f(x) reaches a maximum at m and has points of inflection at m + s.
Bell-shaped curve

13. Chapter 7 – Normal Distribution
Characteristics of the Normal Distribution:
All normal distributions have the same shape but differ in the axis scales.
m = 42.70mm
s = 0.01mm
m = 70
s = 10
Diameters of golf balls
CPA Exam Scores
We can define a standard normal distribution and a transformation to it in order to answer questions about any normal random variable!

14. Chapter 7 – Normal Distribution
Characteristics of the Standard Normal:
Since for every value of m and s, there is a different normal distribution, we transform a normal random variable to a standard normal distribution with m = 0 and s = 1 using the formula:
z =
x – m s
Shift the point of symmetry to zero by subtracting m from x.
Divide by s to scale the distribution to a normal with s = 1.
Denoted N(0,1)

15. Chapter 7 – Normal Distribution
Characteristics of the Standard Normal:
Standard normal PDF f(z) reaches a maximum at 0 and has points of inflection at +1.
Shape is unaffected by the transformation. It is still a bell-shaped curve.
Entire area under the curve is unity.
A common scale from -3 to +3 is used.
The probability of an event P(z1 < Z < z2) is a definite integral of…
However, standard normal tables or Excel functions can be used to find the desired probabilities.

16. Chapter 7 – Normal Distribution
Characteristics of the Standard Normal:
CDF values are tabled and we will use the N(0,1) tables to answer questions about all Normal variables.

17. Chapter 7 – Normal Distribution
Normal Areas from Appendices C-1 & C-2:
Appendix C-1 allows you to find the area under the curve from 0 to z. (Draw on overhead)
Appendix C-2 allows you to find all of the area under the curve left of z. (Hand-out)
Using either of these tables, we can use symmetry and compliments to determine probabilities for the standard normal distribution.

18. Chapter 7 – Normal Distribution
Normal Areas from Appendices C-1 & C-2:
Example: We can use this table to find P(Z < -1.96) and P(Z < 1.96) directly.
P(Z < -1.96) = .025
P(Z < 1.96) = .975

19. Chapter 7 – Normal Distribution
Normal Areas from Appendices C-1 & C-2:
Example: Having found P(Z < -1.96), we can use this result, along with symmetry and the compliment to find several other probabilities…
.9500
P(Z < 1.96) = 1 – P(Z < -1.96)
= = .975
P(Z < -1.96) = .025
P(-1.96 < Z < 1.96)
= P(Z < 1.96) – P(Z < -1.96)
= = .950
Consider P(|Z| > 1.96) = 1 – P(|Z| < 1.96) = 1 – P(-1.96 < Z < 1.96) = 1 – .950 = .050

20. Clickers Use the table from Appendix C-2 (hand-out or
overhead) to determine P(Z < 2.10).
A =
B =
C =
D =
E =

21. Clickers Use the table from Appendix C-2 (hand-out or
overhead) to determine P(Z < -1.20).
A =
B =
C =
D =
E =

22. Clickers Use the table from Appendix C-2 (hand-out or
overhead) to determine P(-1.20 < Z < 2.10).
A =
B =
C =
D =
E =