AP Statistics Chapter 11: Hypoethesis Testing for Means

Prerequisite

AP Statistics Chapter 9 : Hypothesis Testing For Proportions

11.1 Significance Test for Mean

Significance test for mean is testing the claim or the hypothesis about population parameter which is mean $\mu$.

Standardized T Test

Standardized Test Statistic : How far a sample statistic is from the parameter in standardized unit, assuming the null hypothesis $H_0$ is true

\[\begin{gather} T = \frac{\bar{x}- \mu_0}{\displaystyle\frac{s_x}{\sqrt{n}}} \end{gather}\]

Conditions for test

	Observational Study	Experiment
Random	Random Sample	Random Assignment
Independence	10% Condition ($n\leq0.1N$)	Independent Treatments
Normality	Large Counts ($n \geq 30$)	Large Counts ($n \geq 30$)

Paired Data

Paired Data: two different data (values) of the same quantative variable for each individual or each pair of similar individuals.

Instead of doing two sample t test, we can just use the differences from the first place to reduce the work.

Conditions

	Observational Study	Experiment
Random	Random Sample	Random Assignment
Independence	10% Condition ($n_{diff}\leq0.1N$)	Independent Treatments
Normality	Large Counts ($n_{diff} \geq 30$)	Large Counts ($n_{diff} \geq 30$)

11.2 Difference in Mean Test

Hypothesis

Since we are testing if there is difference in means,

$H_0$ : $\mu_1 - \mu_2= 0$

$H_a$ : $\mu_1 - \mu_2 > \text{or} < \text{or} \neq 0$

Conditions

	Observational Study	Experiment
Random	Random Sample	Random Assignment
Independence	10% Condition ($n<0.1N_1$ and $n_2<0.1N_2$)	Independent Treatments
Normality	Large Counts ($n_1 \geq 30$ and $n_2 \geq 30$)	Large Counts ($n_1 \geq 30$ and $n_2 \geq 30$)

In AP, you can plot the graph and see if there is any severe outliers or skew

If not, you can say it is Normal. However, it is not safe in real world.

Standardized T Test

\[\begin{gather} T= \frac{\bar{x}_1 - \bar{x}_2 - (\mu_1-\mu_2)}{\displaystyle\sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}} \end{gather}\]

Four step process of significance test

1. State
   • State Hypothesis and significance level
   • Identify parameter
2. Plan
   • Check conditions
3. Do
   • Calculate
4. Conclude
   • Make conclusion with correct inference and context