What is Hypothesis Testing?

• A method to decide whether data provides enough evidence to reject a claim about a population.
• Uses sample statistics to test statements (hypotheses) about population parameters.

In hypothesis testing, we start with two statements:

\[ H_0: \text{No effect, no difference (status quo)} \]

\[ H_1: \text{Effect or difference exists (research claim)} \]

Example: testing if the mean height differs from 170 cm.

\[ H_0: \mu = 170, \quad H_1: \mu \neq 170 \]

\[ \alpha = P(\text{reject } H_0 \mid H_0 \text{ true}), \quad \beta = P(\text{fail to reject } H_0 \mid H_1 \text{ true}) \]

A smaller α reduces false positives but increases β (false negatives).

1. State $ H_0 $ and $ H_1 $
   
2. Choose significance level $ $
   
3. Compute test statistic:

\[ t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}} \]

1. Find p-value or critical value
   
2. Decide: reject or fail to reject \(H_0\)

Imagine that two classrooms took the same test
- Class A: taught with traditional methods
- Class B: taught with interactive methods

Let us use a t-test to determine if the groups are significantly different
\[ t = \frac{\bar{X_1} - \bar{X_2}}{s_p\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \]
\[ \bar{X_1}, \, \bar{X_2} = \text{sample means} \\ s_p = \text{pooled standard deviation} \\ n_1, \, n_2 = \text{sample sizes} \]

## 
##  Two Sample t-test
## 
## data:  group1 and group2
## t = -3.0399, df = 48, p-value = 0.003825
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -19.990085  -4.073946
## sample estimates:
## mean of x mean of y 
##  99.50005 111.53206

Given a 95% confidence interval and the following:

\[ H_0: \mu_A = \mu_b \qquad H_1: \mu_A \neq \mu_b \]

We can reject the null hypothesis since our p-value < 0.05. We conclude that there is statistically significant evidence that Class B score are higher than Class A with a mean difference of nearly 11 points.