BCOR 1020 Business Statistics

1. BCOR 1020Business 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. 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. Chapter 7 – Continuous Variables Events as Intervals:

4. 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 Xover the range of X. Chapter 7 – Continuous Variables PDFs and CDFs: Normal PDF

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

6. Continuous Cumulative Distribution Functions (CDF’s): Denoted F(x) Shows P(X <x), thecumulative proportion of scores Useful for finding probabilities Chapter 7 – Continuous Variables PDFs and CDFs: 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 balancepoint 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 = 70 s = 10 m = 42.70mm s = 0.01mm 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. x – ms z = 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: • 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. .9500 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… P(Z < -1.96) = .025 P(Z < 1.96) = 1 – P(Z < -1.96) = 1 - .025 = .975 P(-1.96 < Z < 1.96) = P(Z < 1.96) – P(Z < -1.96) = .975 - .025 = .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 = 0.0179 B = 0.1151 C = 0.4821 D = 0.8849 E = 0.9821

21. Clickers Use the table from Appendix C-2 (hand-out or overhead) to determine P(Z < -1.20). A = 0.0179 B = 0.1151 C = 0.4821 D = 0.8849 E = 0.9821

22. Clickers Use the table from Appendix C-2 (hand-out or overhead) to determine P(-1.20 < Z < 2.10). A = 0.0972 B = 0.1151 C = 0.8670 D = 0.8841 E = 0.9821