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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.”