BCOR 1020 Business Statistics Lecture 12 – February 26, 2008

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

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.

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

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

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

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.

Chapter 7 – Continuous Variables Expected Value and Variance:

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.)

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).

Chapter 7 – Normal Distribution Characteristics of the Normal Distribution:

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

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!

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)

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.

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.

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.

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

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) = 1 - .025 = .975 P(Z < -1.96) = .025 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

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

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

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