Topic C9 — Hypothesis Testing
Table of contents
Null hypothesis and alternative hypothesis

Null hypothesis, $H_0$: presumed default state of nature or status quo. In specifying $H_0$, always contains an equality, i.e. $H_0 =$ or $H_0 \leq$ or $H_0 \geq$.

Alternative hypothesis, $H_A$: a contradiction of the default state of nature or status quo. Usually, $H_A$ is the outcome we want to test, the “desired” outcome.
We use sample information to infere unknown population parameters. We do hypothesis testing: we determine whether the evidence from the sample contradicts $H_0$.
Outcomes:
 Reject $H_0$: sample evidence inconsistent with $H_0$
 Do not reject $H_0$: evidence not inconsistent with $H_0$
Importantly, we can never accept $H_0$.
 Twotails: $H_0: \mu = \mu_0$vs $H_A: \mu \neq \mu_0$. If we use $\neq$, we say that both tails matter in order to assess $H_0$. Interpretation of the test: “Has no impact”.
 Onetail: $H_0: \mu \leq \mu_0$vs $H_A: \mu > \mu_0$. Interpretation of the test: “Does not increase”. Look at the sign of $H_A$ to assess which tail to use.
Porcedure
 Identify the relevant population parameter of interest.
 Determine whether it is a one or a twotailed test.
 Include some form of the equality sign in $H_0$ and use $H_A$ to establish a claim.
Hypothesis test using CIs
Decision rule
 Reject $H_0$ if the CI does not contain the value of $H_0=\mu_0$
 Do not reject $H_0$ if CI does contain the value of $H_0 = \mu_0$
Hypothesis test using the pvalue
Assume $H_0$ is true and see if sample evidence contradicts it.
The pvalue is the likelihood of obtaining a sample statistic at least as extreme as the one in the sample, under the assumption that $H_0$ is true.
If the null hypothesis is true, how likely is that I have a sample at least as extreme as the one I have?
 Caluclate the test statistic of your sample.
 Check what is the area associatied with this $z$
 Check if the are is greater (or smaller, depending on the specification of your test) than the area associated with the significance level you choose.
The pvalue tells you the probability that this sample result would be generated if $H_0$ was true. Small pvalue = small probability the null is true
Hypothesis test using the critical value
Reject the null hypothesis if the zscore is greater than some critical value.
 Standardize the sample mean using the sample formula. This is your test statistic.
 Check if it’s greater (or smaller, depending on the type of test) than the $z_{\alpha/2}$ with significance level $\alpha$.
Test statistics for the poplation mean
Same “brick” of formula that you used to compute CI.
Not in the formula sheet! $ \frac{\bar{x}\mu_0}{\sigma/\sqrt{n}} $ or $ \frac{\bar{x}\mu_0}{s/\sqrt{n1}} $ if $\sigma$ unknown.
Test statistics for the population proportion
Not in the formula sheet! $ \frac{\bar{p}p_0}{\sqrt{\frac{p_0(1p_0)}{n}}} $
Type I and Type II errors
 Type I: $H_0$ is true but we believe in $H_A$ (i.e. we don’t believe in $H_0$).

Type II: $H_0$ is not true but we believe in $H_0$ (i.e. we don’t believe in $H_A$).
 Type I: we reject $H_0$ when $H_0$ is true.

Type II: we fail to reject $H_0$ when $H_0$ is false.
 Type I: measured by $\alpha$, significance level
 Type II: measured by $\beta$, power of the test.
Both depend of the standard error.
$H_0$ true  $H_0$ not true  

Reject $H_0$  Type I error False positive $\alpha$  Correct decision Power $1\beta$ 
Do not reject $H_0$  Correct decision Confidence $1\alpha$  Type II error False negative $\beta$ 
A decrease in $\alpha$ will increase $\beta$.
If the null hypothesis is “not pregnant”, the following show type I and II error.