1. Math 145 September 20, 2006

2. Review Methods of Acquiring Data: 1. Census – obtaining information from each individual in the population. 2. Sampling – obtaining information from a part of the population (sample) in order to gain information about the whole population.  Observational Study – observes individuals and measures variables of interest but does not attempt to influence the responses.  Experiments – deliberately imposes some treatment on individuals in order to observe their responses.

3. Example of Designed Experiment Example 1 : Consider the problem of comparing the effectiveness of 3 kinds of diets (A, B, C). Forty males and 80 females were included in the study and were randomly divided into 3 groups of 40 people each. Then a different diet is assigned to each group. The body weights of these 120 people were measured before and after the study period of 8 weeks and the differences were computed. Example 2 : In a classic study, described by F. Yates in the The Design and Analysis of Factorial Experiments, the effect on oat yield was compared for three different varieties of oats (A, B, C) and four different concentrations of manure (0, 0.2, 0.4, and 0.6 cwt per acre).

4. Terminologies in Experiments  Experimental Units – These are the individuals on which the experiment is done.  Subjects – human beings.  Response variables – Measurement of interest.  Factors – Things that might affect the response variable (explanatory variables). {new drug}  Levels of a factor – {different concentration of the new drug; no drug, 10 mg, 25 mg, etc.}  Treatment – A combination of levels of factors.  Repetition – putting more than 1 experimental units in a treatment.

5. Example 1 : Diet Study Example 1 : Consider the problem of comparing the effectiveness of 3 kinds of diets (A, B, C). Forty males and 80 females were included in the study and were randomly divided into 3 groups of 40 people each. Then a different diet is assigned to each group. The body weights of these 120 people were measured before and after the study period of 8 weeks and the differences were computed. a)Experimental units : People b)Response variable : Weight lost c)Factor(s) : Diet d)Levels : diet A, diet B, diet C e)Treatments : diet A, diet B, diet C

6. Example 2 : Oat Yield Study Example 2 : In a classic study, described by F. Yates in the The Design and Analysis of Factorial Experiments, the effect on oat yield was compared for three different varieties of oats (A, B, C) and four different concentrations of manure (0, 0.2, 0.4, and 0.6 cwt per acre). a)Experimental units : Fields b)Response variable : Oat yield c)Factor(s) : Oat variety, Manure concentration d)Levels : Oat A, B, C ; Concentration 0,.2,.4,.6 e)Treatments : (A, 0), (A,.2), …, (C,.6)

7. Designs of Experiments  Completely Randomized – Experimental units are allocated at random among all treatments.  Double-Blind Study – Neither the subjects nor the medical personnel know which treatment is being giving to the subject.  Matched Pair – Used for studies with 2 treatment arms, where an individual from one group is matched to another in the other group.  Block Design – The random assignment of units to treatments is carried out separately within each block.  Block – is a group of experimental units that are known to be similar in some way that is expected to affect the response to the treatment.

8. Example 1 : Diet Study Example 1 : Consider the problem of comparing the effectiveness of 3 kinds of diets (A, B, C). Forty males and 80 females were included in the study and were randomly divided into 3 groups of 40 people each. Then a different diet is assigned to each group. The body weights of these 120 people were measured before and after the study period of 8 weeks and the differences were computed.  Block - Gender

9. Section 3.3 : Sampling Designs  Simple Random Sampling.  Systematic Sampling.  Cluster Sampling.  Stratified Sampling (with proportional Allocation).

10. Section 3.4 : Sampling Distributions Parameters vs. Statistics.  Parameter – a number that describes the population. A parameter is a fixed number, but in practice we do not know its value.  Statistic – a number that describes a sample. We often use a statistic to estimate the value of an unknown parameter. It’s value changes from sample to sample.  Sampling Distribution  A Statistic used to estimate a parameter is unbiased if the mean of its sampling distribution is equal to the parameter value.

11. Homework Exercises: Sec 3.1 & 3.2: #1, 2, 3, 5, 9, 11, 12, 20*, 21, 23. Sec 3.3 & 3.4: 36, 37, 38, 52, 56, 57, 63, 64, 66, 76*.

12. Thank you!