Hypothesis testing is a formal statistical framework used to make inferences about population parameters based on sample data.

• Null Hypothesis (\(H_0\)): The baseline statement claiming no effect , no change, or status quo.
• Alternative Hypothesis (\(H_a\)): The statement reflecting the claim we seek evidence to support.

We gather a sample to evaluate if the evidence is strong enough to reject \(H_0\).

Consider a standar one-sample \(z\)-test evaluating a population mean \(\mu\) against a specific baseline value \(\mu_0\):

\[H_0 \mu = \mu_0\] \[H_a: \mu \neq \mu_0\]

Assuming standard conditions are met, our test statistic follows a standard normal distribution under the null hypothesis:

\[Z = \frac{\bar{X}$ - \mu_0}{\sigma/ \sqrt{n}} \sim N(0, 1) \]

Where \(\bar{X}\) is the sample mean, \(\sigman\) is the population standard deviation, and \(n\) is the sample size.

We evaluate our calculated test statistics against a predefined threshold:

• Significance Level (\(\alpha\)): The probability of committing a Type I Error (rejecting a true \(H_0\)). Typically set to \(0.05\). _ The \(p\)-value: The probability of obtaining a test statistic at least as extreme as the observed value, assuming \(H_0\) is true.

\[\text{If } p\text{-value} \leq \alpha \implies \text{Reject } H_0\] \[\text{If } p\text{-value} > \alpha \implies \text{Fail to Reject } H_0\]

Let’s test if a sample of exam scores deviates significantly from a historical mean score of \(\mu_0 = 70\).

## 
##  One Sample t-test
## 
## data:  scores
## t = 2.2373, df = 49, p-value = 0.02985
## alternative hypothesis: true mean is not equal to 70
## 95 percent confidence interval:
##  70.37082 76.91574
## sample estimates:
## mean of x 
##  73.64328

Here is a look at our sample distribution with a reference line marking the historical population mean (\(\mu_0 = 70\)).

This plot maps a theoretical normal sampling distribution curve, highlighting the boundaries of our critical regions.

This interactive jitter plot allows users to hover over individual sample scores to view their specific data values.