2086 Lecture 5 Hypothesis Testing

Lecture Note: Lecture 5 Notes.pdf

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Hypothesis testing

Hypothesis testing includes two hypothesis, null hypothesis and alternative hypothesis. Alternative hypothesis is the against conclusion of null hypothesis. After we decided our two hypothesis, we will calculate a p-value, the formula of p-value depends on our hypothesis:

$$p = \begin{cases} 2\mathbb{P}(Z < -|z_{\hat{\mu}}|) & \text{if } H_0 : \mu = \mu_0 \ \text{vs}\ H_A : \mu \neq \mu_0, \\[1em] 1 - \mathbb{P}(Z < z_{\hat{\mu}}) & \text{if } H_0 : \mu \leq \mu_0 \ \text{vs}\ H_A : \mu > \mu_0, \\[1em] \mathbb{P}(Z < z_{\hat{\mu}}) & \text{if } H_0 : \mu \geq \mu_0 \ \text{vs}\ H_A : \mu < \mu_0. \end{cases}$$

Notice that $\mu$ here represent the actual population parameter, and the $\mu_0$ represent the hypothesis parameter.

We can calculate $z_{\hat{\mu}}$ by

$$z_{\hat{\mu}} = \frac{\hat{\mu} - \mu_0}{\sqrt{\sigma^2 / n}}$$

We don’t know the population parameter, so we use the sample parameter, and than find the difference between sample parameter and hypothesis parameter, divide by standard deviation so we know how many standard deviation is between sample and hypothesis, and then we can use z-table to find the p-value.

If the p-value > 0.1 means we do not have enough evidence against the null hypothesis, because there will be more then 10% chance of seeing such a result

If p-value is between 0.1 and 0.05, we have a marginal evidence against the null hypothesis

If p-value is between 0.05 and 0.01, we have a quite strong evidence against the null hypothesis

If p-value is even lower, we have a strong evidence against the null hypothesis, if we take 100 different samples, only one of them will have such estimated parameter