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A hypothesis test is a statistical method used to decide whether there is enough evidence in a sample of data to support a particular belief or claim about a population.

The Two Hypotheses

Every test starts with two competing statements:

• Null Hypothesis ($H_0$): The “default” position. It assumes nothing has changed or the original claim is still true. It always uses an equals sign (e.g., $p = 0.5$).
• Alternative Hypothesis ($H_1$): The new claim you are investigating. This is what you suspect might actually be happening.

Test Types (One-tailed vs. Two-tailed)

The “tail” of a test depends on what you are looking for in the alternative hypothesis ($H_1$).

Note

In a two-tailed test, you must split the significance level in half (e.g., a $5\%$ test becomes $2.5\%$ at each end).

Key Vocabulary

• Test Statistic: The piece of data you are looking at (e.g., the number of people who liked a product in a trial).
• Significance Level ($\alpha$): The “threshold” of probability. It is the maximum risk you are willing to take of being wrong. Common levels are $5\%$ ($0.05$) or $1\%$ ($0.01$).
• Critical Value: The “cut-off” point that separates the acceptance region from the critical region.
• Critical Region (Range): The “rejection zone.” If your result falls here, it is so unlikely to happen by chance that you reject the null hypothesis.
• Acceptance Region: The “safe zone.” If your result falls here, you do not have enough evidence to change your mind, so you keep the null hypothesis.

The 5-Step Process

1. State the Hypotheses: Write down $H_0$ and $H_1$ clearly using the correct symbols.
2. Identify the Model: Usually a Binomial Distribution $B(n, p)$ in Year 1.
3. Find the Probability: Calculate the probability of getting your result (or one more extreme) assuming $H_0$ is true.
4. Compare: Compare your probability ($p$-value) to the significance level.
5. Conclude:
   • If $p <$ Significance Level: Reject $H_0$. The result is significant.
   • If $p >$ Significance Level: Do not reject $H_0$. The result is not significant.

Final Conclusion Logic

When writing your final answer, you must provide two parts:

1. Statistical result: “Reject $H_0$” or “Fail to reject $H_0$.”
2. Contextual result: “There is sufficient evidence to suggest that the new claim is true” or “There is insufficient evidence to suggest the new claim.”

Example

“The result is in the critical region, therefore we reject $H_0$. There is evidence at the 5% level to suggest the new coin is biased.”