Eleisha's Segment 12: P-Value Tests

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 ${\text{Student}}(0,\sigma ,4)$? 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 ${\text{Student}}(0,\sigma ,4)$, then for a two sided test the critical region is when $|t|\approx 2.7765\sigma$. 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 $\mu$ (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 $\lambda$ is:

$p(x)=\lambda e^{-\lambda x}$

${\text{Since the mean of p(x) is }}{\frac {1}{\lambda }},{\text{we take }}\lambda ={\frac {1}{\mu }}$ 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