1 STAT 6020 Introduction to Biostatistics Fall 2005 Dr. G. H. Rowell Class 2

2 Review Questions  Chapter 1 Statistical Inference  Chapter 2 Data Types: Numerical/Categorical  Chapter 3 What is the difference in a bar chart & a histogram? Describe a useful transformation & how it works.

3 Ch4: Theoretical Distributions, An Overview  Probability  Samples/Population  Distributions Continuous  Normal, Lognormal, Uniform Discrete  Binomial, Poisson

4 Ch 4: Probability  We teach an entire course on this – STAT 6160  Not a main focus of this course  Understand Basic Axioms Randomness Independence Probability Distributions Functions

5 Ch 4: Probability - Basics S = Sample space E = an event in the Sample Space P(E) = Probability that event E occurs 0<= P(E) <=1 P(S) = 1 If E1, E2, E3, … are mutually exclusive events, then probability of the union of events = sum of the individual events P(E1 U E2 U E3 U …) = P(E1) + P(E2) + P(E3) + … for a finite or an infinitely countable number of events

6 Ch 4: Probability - Independence  Independent Events Events A & B are independent if and only if P(A given that you know everything about B) = P(A) OR P(A and B) = P(A) * P(B) Over simplifying: A & B are independent if knowing the outcome of A tells us nothing about B

7 Ch 4: Sample & Populations  Population  Sample  Goal of Statistics

8 Ch 4: Probability Distributions  Decision: Continuous or Discrete ?  If Continuous, what is the shape of the relative frequency of the outcomes? Flat – Uniform Bellshaped – Normal Positively Skewed – Lognormal

9 Ch 4: Probability Distributions  Decision: Continuous or Discrete ?  If Continuous, what is the shape of the relative frequency of the outcomes? Flat – Uniform Bellshaped – Normal Positively Skewed – Lognormal

10 Ch 4: Probability Distributions  If Discrete, what experiment is the variable modeling Counts number of successes – might be binomial Counts number of trials to the first success – might be geometric Counts independent, random, and RARE events – might be Poisson

11 Ch 4: Normal Distribution  Mound-shaped and symmetrical  Mean and standard deviation used to describe the distribution  “Empirical Rule”

12 Standard Normal  Normal with mean zero and standard deviation 1 Notation: N(0, 1)  Z-score Formula Meaning  Tools for finding probabilities Tables, software, applets

13 Statistical Software Online  StatCrunch  StatiCui  VassarStats

14 Visualization  What does “normal” look like? Histogram: See Figure 4.7, page 60.  Normal Density Function  Normal Cumulative Distribution

15 Ch 4: Example, Normal  If the average daily energy intake of healthy women is normally distributed with a mean of 6754 kJ and a standard deviation of 1142 kJ than what is the probability that a randomly selected women is below the recommended intake level of 7725 kJ per day? Above 7725 kJ? Between 6000 and 7000 kJ?

16 Ch 4: Serum Albumin Example  Data: 216 patients with primary biliary cirrhosis mean serum albumin level: g/l, st dev = 5.84 g/l See histogram, Fig 4.5 page 56, follows normal distribution  Constructing Chart on Page

17 Ch 4: A Continuous Skewed Right Distribution: Lognormal  Example: Serum Bilirubin, page 61

18 Ch 4: Continuous Distribution: Uniform  Conditions for Uniform  Visualization

19 Ch 4: Discrete Distributions Binomial Distribution  Binomial Experiment:  Binomial Random Variable:  Binomial Distribution Function:

20 Ch. 4: Binomial Example

21 Ch. 4: Binomial Visualization  Homework: Complete the Binomial Visualization Activity found at 1/Pages1/Home.htm Be sure to submit the “Pretest” and the “Lesson.” You may want to print the results as a back-up. This is a Hand-in Homework worth 10 points.

22 Ch 4: Discrete Distributions: Poisson Distribution  Conditions for a Poisson Distribution:  Poisson Visualization: /applets/PoiDensityApplet.html

23 Ch 4: Homework  Exercises # 1 – 8 Check you answers in the Back of the book.  Bring to class for next week – the mean and standard deviation for heights of Americans of your gender.