1. Experiments, Simulations Confidence Intervals
Week 4

2. Terms Individuals in an experiment are called experimental units.
Humans experimental units are called subjects or participants.
The values a factor can take on are call levels.
All the levels of all the factors that one experimental unit receives is called its treatment.

3. Four Principles of Experimental Design
Control:
Control the sources of variation not coming from the factors.
Attempt to make the groups as alike as possible except for the treatments.
Randomize:
Randomization allows us to equalize unknown or uncontrollable sources of variation.
Random assigning of subjects to treatment groups helps neutralize lurking variables.

4. Four Principles of Experimental Design
Replication:
Doing the experiment on a sufficient number of subjects. The more the better.
Being able to duplicate the entire experiment and get similar results.
Block:
Used to help eliminate the effect of know traits of the subjects that might affect the results.
Block into homogeneous groups and randomly assign treatments within the block.

5. Blocking
Put experimental units into like groups by a possible explanatory variable.
Randomly assign treatments inside of each block
Analyze the difference between the treatments within each block.
This method removes that explanatory variable as a possible lurking variable

6. Statistical Significance
There will be differences between the treatment groups. It is to be expected.
If the differences are larger than what we would expect from randomness then we say the differences are Statistically Significant

7. Control Group
Often in an experiment there is a group that gets no treatment. This is treatment group is called a Control Group.
A control group is used to get a “baseline” to be used as a comparison

8. Blindness
If the subjects in an experiment are not aware of what treatment group they are in the experiment is said to be single blind.
If both the subjects and the people applying the treatment do not know which treatment group a subject is in the experiment is said to be double blind.

9. Placebo/Placebo Effect
A placebo is a fake treatment.
Often a subject will show improvement even with a fake treatment. This improvement for no apparent reason is called the placebo effect.
This is often due the emotional effect of being a part of a study.

10. Completely Randomized Experiment
A completely randomized experiment is one in which each experimental unit has an equal chance of getting any of the treatments

11. Probability Exploration Tool
Simulations
Probability Exploration Tool

12. Simulations A simulation is made up of
A event called a component that is repeated.
Each component has a finite collection of results that happen at random.
A trial is the occurrence of one component.

13. Creating a Simulation Identify the component to be repeated
Explain how you will assign the random numbers to accurately simulate the distribution of results.
Explain how you will conduct a trial.
State the response variable.
Run the trials.
Analyze the results
State your conclusion.

14. Assigning the Numbers How many digits do you need?
What numbers will be a success?
What numbers will be a failure?
Are there numbers you will ignore?
Do the percentages work out?

15. Confidence Intervals for Proportions

16. Standard Error (SE)
When we do not know the standard deviation of the population and estimate it using the standard deviation of the sample we call it the Standard Error.

17. Confidence Intervals have:
A level of confidence
Usually 90%, 95% or 99%
Usually supplied by an outside source
An interval
Usually constructed from a sample
Sample estimate ± margin for error

18. Confidence Interval Info
The wider the interval the more confident that it captures the true proportion.
I can be 100% sure that the proportion of left-handed AP Stats students is between 0 and 1.
If you want a narrow interval you must settle for less confidence that the population proportion is in there.

19. Creating Confidence Intervals
Estimate ± Margin for Error
Estimate will be the sample proportion p-hat
Our margin for error (ME) depends on the level of confidence required.
If we went 2 standard deviations either side by the rule we should be 95% sure we captured the population proportion. If we went out 1 standard deviation we should be 68% sure we captured the population proportion

20. Confidence Interval z numbers
z* (Critical value) gives the number of standard errors away from the sample proportion need for that level of confidence
Common z* values
90% ==> 1.645
95% ==> 1.96
99% ==> 2.576

21. Assumptions and Conditions
Independence
Logical - it would make sense to be independent
Random - the sample is randomly chosen
10% condition
Large enough sample

22. How to do it? If all assumptions and conditions are met: Find
Find the correct z*
Construct the interval

23. Calculating Needed Sample Size
Often you will be given the level of confidence AND a desired margin of error and asked for a confidence interval.
How can we do both?
and
So if we set ME = the desired margin of error and solve for n we should get the need sample size.

24. Calculating Needed Sample Size
But wait a minute! We are using the sample proportion to find the sample size, before we take the sample!!!
What do we use for the sample proportion?
Two answers;
The results of previous research.(Stat Fairy again)
Use This gives an n that can handle the worse case possiblilities.