Last Time (Sampling &) Estimation Confidence Intervals Started Hypothesis Testing

CONFIDENCE INTERVAL FOR  X WHEN  X KNOWN 100(1 -  )% confidence interval ± · P( Confidence Interval encloses  X ) = 1 -  Note:  = P(Confidence Interval misses  X ) Note the relationship between:  = probability of missing  X n = sample size = half-width of the confidence interval = margin of error Fix any two and the third is determined! XX

CONFIDENCE INTERVAL FOR  X WHEN  X KNOWN 100(1 -  )% confidence interval ± · Margin of Error Confidence Intervals are wide if population variance of X is large sample size is small Wide CI means that we are not very precise in estimating the population parameter (here the population mean)

Caution! Confidence interval:

CONFIDENCE INTERVAL FOR  X WHEN  X KNOWN

N(0,1)

CONFIDENCE INTERVAL FOR  X WHEN  X KNOWN N(0,1)

CONFIDENCE INTERVAL FOR  X WHEN  X KNOWN N(0,1)

Confidence interval for  X

Example: Let’s consider once more the age of entering students in U.S. B-schools. Suppose (for simplicity) that  X = 3. An old government publication says the average age is  X = 28. You think that it is higher and you want to check your intuition statistically. You collect a random sample of size n=330. It turns out in the sample the average age is 28.5.

Example: Claim of the Government Publication:  X = 28 Null Hypothesis: H 0 :  X = 28 (Status Quo, Common Wisdom) Alternative Hypothesis: H a :  X > 28 (Your claim) Rules of the game: Unless you can provide a lot of evidence for your claim, the status quo will prevail. The null hypothesis will be retained until enough evidence has been accumulated against it. In particular, all calculations will assume that  X = 28. (!!!)

 H a is the hypothesis you are gathering evidence in support of.  H 0 is the fallback option = the hypothesis you would like to reject.  Reject H 0 only when there is lots of evidence against it.  A technicality: always include “=” in H 0  H 0 (with = sign) is assumed in all mathematical calculations!!! Innocent Guilty

In our Example: Recall: Suppose we know that  X = 3. Suppose that  X = 28. What is the probability that is more than 28.5? (n=330) $1,000,000 Question: How much evidence do we have against H o ? Assuming H o, how unusual is the sample mean 28.5?

Suppose we know that  X = 3. H o :  X = 28, H a :  X > 28 What is the probability that is more than 28.5? (n=330) Given a population mean of 28, the probability of a sample mean further to the right than 28.5 is.0012 The p-value of this hypothesis test is.0012: Given that the Null Hypothesis holds, it is extremely unlikely to observe a sample mean equal or larger the one we obtained.  REJECT H o

Hypothesis Testing p-value

Hypothesis Testing p-value

Hypothesis Testing p-value

We will get back to hypothesis testing using p-values towards the end of class. Before that, let’s think once more about our legal system and the “innocent until proven guilty” rule. We will talk about three different methods of conducting hypothesis tests.

Today More on hypothesis testing…

Great! Type II Error Great! Type I Error Four scenarios when making a decision based on a sample

Recall:  H A is the hypothesis you are gathering evidence in support of.  H 0 is the fallback option = the hypothesis you would like to reject.  Reject H 0 only when there is lots of evidence against it. Specify the probability of Type I error you are ready to tolerate. (  = significance level of the test) Reject H 0 if, P(observing something more extreme than what you observed)  

Example

100

Example

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Example Alternatively, we could compute 95% Confidence Interval for  :

Example Would the observed evidence (sample mean =100.5) be sufficient to reject the null hypothesis at:   =.1 significance level?   =.02 significance level?   =.01 significance level?   =.001 significance level? What is the smallest  for which the observed evidence would be sufficient for rejecting H 0 ?

100 Computing a p-value for the sample mean Shaded area is p-value of observed sample mean Given null hypothesis, how likely am I to observe a sample mean more extreme than the one I actually observed?

100 Computing a p-value for the sample mean Shaded area is p-value of observed sample mean Given null hypothesis, how likely am I to observe a sample mean more extreme than the one I actually observed?

How to compute (two-sided) p-value: Example

P-value

Three equivalent methods of hypothesis testing (  =significance level)

P-value (the amount of doubt we can have) “Innocent until proven guilty beyond a reasonable doubt”. How much doubt is “a reasonable doubt”? Is there reasonable doubt if 5% or fewer of innocent people show the behavior that the accused showed? P-value: Among innocent people, only a proportion p display the behavior that the accused has displayed. If p is very small, then the accused is proven guilty beyond a reasonable doubt.

P-value

Hypothesis Testing using Critical Values N(0,1)

Hypothesis Testing using Critical Values N(0,1)

Hypothesis Testing using Critical Values N(0,1)