Hypothesis Tests

Definitions

In a hypothesis test, you take a claim and test it against statistical evidence.
The claim that you’re testing is called the null hypothesis test. It’s represented as $H_{0}$ , and it’s the claim that’s accepted unless there’s strong statistical evidence against it.
The alternate hypothesis is the claim we’ll accept if there’s strong enough evidence against $H_{0}$ . It’s represented by $H_{1}$ .
The test statistic is the statistic you use to test your hypothesis. It’s the statistic that’s most relevant to the test. You choose the test statistic by assuming that $H_{0}$ is true.
The significance level is represented by $α$ . It’s a way of saying how unlikely you want your results to be before you’ll reject $H_{0}$
The critical region is the set of values that presents the most extreme evidence against the null hypothesis test. You choose your critical region by considering the significance level and how many tails you need to use.
A one-tailed test is when your critical region lies in either the upper or the lower tail of the data ( $α$ ). A two-tailed test is when it’s split over both ends ( $α / 2$ ). You choose your tail by looking at your alternate hypothesis.
A p-value is the probability of getting the result of your sample, or a result more extreme in the direction of your critical region.
If the p-value lies in the critical region, you have sufficient reason to reject your null hypothesis. If your p-value lies outside your critical region, you have insufficient evidence.

Testing Steps

Decide on the hypothesis you’re going to test
Choose your test statistic
Determine the critical region for your decision
Find the p-value of the test statistic
See whether the sample result is within the critical region
Make your decision

Example

drug company claims a drug cures 90% of patients in 2 weeks. Here's a doctor's data:

Cured?	Yes	No
Frequency	11	4

$H_{0} : p = 0.9$ , $H_{1} : p < 0.9$
$X \sim B (15, 0.9)$
$P (X < c < α)$ where $α = 5$ %
$P (X \leq 11) = 1 - P (X \geq 12) = 0.0555$
$P (X \leq 11) > α$
therefore we fail to reject $H_{0}$

Errors

A Type I error is when you reject the null hypothesis when it’s actually correct. The probability of getting a Type I error is $α$ , the significance level of the test.
A Type II error is when you accept the null hypothesis when it’s wrong. The probability of getting a Type II error is represented by $β$ .
To find $β$ , your alternate hypothesis must have a specific value. You then find the range of values outside the critical region of your test, and then find the probability of getting this range of values under $H_{1}$ .

	Accept $H_{0}$	Reject $H_{0}$
$H_{0}$ True	✅	Type I error
$H_{0}$ False	Type II error	✅

The power of a hypothesis test is the probability that we will reject $H_{0}$ when $H_{0}$ is false. In other words, it’s the probability that we will make the correct decision to reject $H_{0}$ .

P o w e r = 1 - β

Example

drug company claims a drug cures 90% of patients in 2 weeks. Here's a doctor's data:

Cured?	Yes	No
Frequency	80	20

$P (T y p e I e r r o r) = 0.05$
$H_{0} : p = 0.9$ , $H_{1} : p = 0.8$
Find values outside critical region: $P (Z < - 1.64) = 0.05 \to Z \geq - 1.64$
De-standardize to find $X$ assuming $H_{0}$ is true where $X \sim N (90, 9)$ : $$\frac{X-90}{3}\geq -1.64\rightarrow X\geq 85.08$$
Find $P (X \geq 85.08)$ , assuming that $H_{1}$ is true where $X \sim N (80, 16)$ :

z = \frac{85.08 - 80}{\sqrt{16}} = 1.27

P (Z \geq 1.27) = 1 - P (Z < 1.27) = 0.102

P (T y p e I I e r r o r) = 0.102

P o w e r = 1 - P (T y p e I I e r r o r) = 0.898

This means that the probability that we will make the correct decision to reject the null hypothesis is 0.898

$χ^{2}$ Distribution

Use Cases

Two Variables
Goodness of Fit or Independence

Steps

Decide on the hypothesis you’re going to test, and its alternative
Find the expected frequencies and the degrees of freedom
Determine the critical region for your decision
Calculate the test statistic $X^{2}$
See whether the test statistic is within the critical region
Make your decision

Distribution	Condition	$ν$
Binomial	You know what $p$ is	$$\nu=n-1$$
Binomial	You don’t know what $p$ is, and you have to estimate it from the observed frequencies	$$\nu=n-2$$
Poisson	You know what $λ$ is	$$\nu=n-1$$
Poisson	You don’t know what $λ$ is, and you have to estimate it from the observed frequencies	$$\nu=n-2$$
Normal	You know what $μ$ and $σ^{2}$ are	$$\nu=n-1$$
Normal	You don’t know what $μ$ and $σ^{2}$ are, and you have to estimate them from the observed frequencies	$$\nu=n-3$$
Sources: 1

Definitions

Testing Steps

Example

Errors

Example

χ2 Distribution

Use Cases

Steps

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$χ^{2}$ Distribution