STAT10x10.6.05HypothesisTests

# STAT10x10.6.05HypothesisTests - Hypothesis Testing Like...

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Hypothesis Testing Like proof by contradiction : (think back to geometry) Example : if x and y are two even numbers, prove that x+y is also even. Proof (by contradiction). Assume x+y is an odd number. Than x+y = 2c+1 for some integer c. If x and y are both even, then x=2a and y=2b for some integers a and b. Then we have (odd number) 2c+1 = 2a + 2b = 2(a+b) (even number) Therefore, x+y must be even. Math, Logic Statistics, Real Life Want to prove a statement. Assume the statement is not true. Reason to a contradiction. Want to use data to provide evidence for a hypothesis. Assume the opposite hypothesis is true. Show the observed data is very unlikely STAT 101-106a Introduction to Statistics 191

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(i.e. prove something is impossible) (not impossible, just very unlikely) STAT 101-106a Introduction to Statistics 192
Example : IQ Tests . The general population has a mean IQ score of 100 and the population standard deviation of scores is 16 = σ . Suppose a sample of 9 CIA operatives yields a sample mean IQ score of n X =126. We’re wondering if on average, CIA operatives are smarter than the general population. “Null Hypothesis” H 0 : 100 = μ “Alternative Hypothesis” H a : 100 We want to assess evidence for H a . Assume the opposite; that is, assume H o is true!! If the null hypothesis is true, then (according to the Central Limit Theorem), our sample mean n X should look like an observation from a normal distribution with mean 100 and standard deviation 16/sqrt(9) : n X ~ ) 3 16 , 100 ( N STAT 101-106a Introduction to Statistics 193 Read as ‘H naught’

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How likely is it to observe a value of 126 or larger in a normal distribution with mean 100 and standard deviation 16/3? That is, what is ) 126 ( n X P This is a problem we know how to solve!! Calculate z-score : 875 . 4 3 16 100 126 = - (or just look up probability in MINITAB) In General : Calculate z-score as n x o n σ μ - z-score = What is the probability of seeing a value of 4.875 of greater in a standard normal distribution? Basically ZERO!!!!! 000 . 0 ) 875 . 4 ( = n Z P p-value STAT 101-106a Introduction to Statistics 194 Sample mean – true mean under null hypothesis True standard deviation / sqrt(sample size)
The p-value is the probability of observing our sample mean or something more extreme IF the null hypothesis is true. For the CIA IQ data, we REJECT THE NULL HYPOTHESIS because the p-value is quite SMALL. How small is small? Reject the null hypothesis when p-value is less than a pre-specified threshold This threshold is called α (alpha) The generally accepted minimum threshold is =.05 If the p-value is not less than , we FAIL TO REJECT the null hypothesis STAT 101-106a Introduction to Statistics 195 Memorize This!

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## This note was uploaded on 04/07/2008 for the course STAT 102 taught by Professor Jonathanreuning-schererdonaldgreen during the Fall '05 term at Yale.

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STAT10x10.6.05HypothesisTests - Hypothesis Testing Like...

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