# Difference between revisions of "Eleisha's Segment 12: P-Value Tests"

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1. What is the critical region for a 5% two-sided test if, under the null hypothesis, the test statistic is distributed as <math> \text{Student}(0,\sigma,4) </math>? That is, what values of the test statistic disprove the null hypothesis with p < 0.05? (OK to use Python, MATLAB, or Mathematica.) | 1. What is the critical region for a 5% two-sided test if, under the null hypothesis, the test statistic is distributed as <math> \text{Student}(0,\sigma,4) </math>? That is, what values of the test statistic disprove the null hypothesis with p < 0.05? (OK to use Python, MATLAB, or Mathematica.) | ||

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+ | Let t = the value of the test statistic | ||

+ | If the test statistic is distributed as <math> \text{Student}(0,\sigma,4) </math>, then for a two sided test the critical region is when <math> |t| \approx 2.7765 \sigma </math>. This can calculated by taking the inverse of the CDF of the probability distribution and and evaluating it at (1 - 0.05/2). | ||

2. For an exponentially distributed test statistic with mean <math>\mu </math> (under the null hypothesis), when is the the null hypothesis disproved with p < 0.01 for a one-sided test? for a two-sided test? | 2. For an exponentially distributed test statistic with mean <math>\mu </math> (under the null hypothesis), when is the the null hypothesis disproved with p < 0.01 for a one-sided test? for a two-sided test? | ||

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+ | Let t = the value of the test statistic | ||

+ | The pdf for an exponentially distributed test statistic with parameter <math>\lambda </math> is: | ||

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+ | <math> p(x) = \lambda e^{- \lambda x} </math> | ||

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+ | <math> \text{Since the mean of p(x) is } \frac{1}{\lambda}, \text{we take } \lambda = \frac{1}{\mu} </math> | ||

+ | We can solve for the critical region in a similar manner to question one by determining the inverse of the CDF of p(x) and evaluating it a (1 - 0.01) for a one sided test and (1 - .01/2). | ||

+ | For a one -sided test the null hypothesis is disproved with p< 0.01 when t> 4.60517. For a one -sided test the null hypothesis is disproved with p< 0.01 when |t|> 4.60517. | ||

## Revision as of 18:52, 22 February 2014

** To Calculate: **

1. What is the critical region for a 5% two-sided test if, under the null hypothesis, the test statistic is distributed as ? That is, what values of the test statistic disprove the null hypothesis with p < 0.05? (OK to use Python, MATLAB, or Mathematica.)

Let t = the value of the test statistic If the test statistic is distributed as , then for a two sided test the critical region is when . This can calculated by taking the inverse of the CDF of the probability distribution and and evaluating it at (1 - 0.05/2).

2. For an exponentially distributed test statistic with mean (under the null hypothesis), when is the the null hypothesis disproved with p < 0.01 for a one-sided test? for a two-sided test?

Let t = the value of the test statistic The pdf for an exponentially distributed test statistic with parameter is:

We can solve for the critical region in a similar manner to question one by determining the inverse of the CDF of p(x) and evaluating it a (1 - 0.01) for a one sided test and (1 - .01/2). For a one -sided test the null hypothesis is disproved with p< 0.01 when t> 4.60517. For a one -sided test the null hypothesis is disproved with p< 0.01 when |t|> 4.60517.

** To Think About: **

1. P-value tests require an initial choice of a test statistic. What goes wrong if you choose a poor test statistic? What would make it poor?

2. If the null hypothesis is that a coin is fair, and you record the results of N flips, what is a good test statistic? Are there any other possible test statistics?

3. Why is it so hard for a Bayesian to do something as simple as, given some data, disproving a null hypothesis? Can't she just compute a Bayes odds ratio, P(null hypothesis is true)/P(null hypothesis is false) and derive a probability that the null hypothesis is true?

** Back To: ** Eleisha Jackson