variance formula. probability a. the importance of probability hypothesis testing and statistical...
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Probability
A. The importance of probability
Hypothesis testing and statistical significanceProbabilistic causation - because error always exists
in our sampling (sampling error) we can only deal with probabilities of being correct or incorrect in our conclusions
The Normal Curve
A. What is a Probability Distribution
-How does it differ from a freq. distribution?-Theory vs. Empirical Observations-Examples?
B. The normal curve represents a probability distribution
-Why does it make sense to use the idea of a normal curve?-The area under the curve-Theory v. Reality
The Normal Curve
• The mean and standard deviation, in conjunction with the normal curve allow for more sophisticated description of the data and (as we see later) statistical analysis
• For example, a school is not that interested in the raw GRE score, it is interested in how you score relative to others.
• Even if the school knows the average (mean) GRE score, your raw score still doesn’t tell them much, since in a perfectly normal distribution, 50% of people will score higher than the mean.
• This is where the standard deviation is so helpful. It helps interpret raw scores and understand the likelihood of a score.
• So if I told you if I scored 710 on the quantitative section and the mean score is 591. Is that good?
• It’s above average, but who cares.
• What if I tell you the standard deviation is 148?
• What does that mean?
• What if I said the standard deviation is 5?
• Calculating z-scores
z-scores & conversions
• What is a z-score?– A measure of an observation’s distance from
the mean.– The distance is measured in standard
deviation units.• If a z-score is zero, it’s on the mean.• If a z-score is positive, it’s above the mean.• If a z-score is negative, it’s below the mean.• If a z-score is 1, it’s 1 SD above the mean.• If a z-score is –2, it’s 2 SDs below the mean.
Converting raw scores to z scores
What is a z score? What does it represent
Z = (x-µ) / σ
Z = (710-563)/140 = 147/140 = 1.05
Converting z scores into raw scores
X = z σ + µ [(1.05*140)+563=710]
Examples of computing z-scores
5 3 2 2 1
6 3 3 2 1.5
5 10 -5 4 -1.25
6 3 3 4 .75
4 8 -4 2 -2
X X XX SD SD
XXz
11
The Normal Curve
• A mathematical model or and an idealized conception of the form a distribution might have taken under certain circumstances.
– A sample of means from any distribution has a normal distribution (Central Limit Theorem)
– Many observations (height of adults, weight of children in Nevada, intelligence) have Normal distributions
Finding Probabilities under the Normal Curve
So what % of GRE takers scored above and below 710? (Z = 1.05)
The importance of Table A in Appendix C
- Why is this important?Infer the likelihood of a result
Confidence Intervals/Margin of Error
Inferential Statistics (to be cont.- ch.6-7)
MIDTERM (OCT. 6th)What’s on it?
Babbie Chapters 1,2,4,5,6Fox/Levin Chapters 1,2,3,4,5Lecture Notes Through Oct. 4th
Statistical Calculations?Calculate the mean, median, modeCalculate the Variance & Standard Deviation
Extra credit: calculate a z-score/using Table A
Computer/SPSS questions? NO