Hypothesis Testing

1-Tail Test Example

Null Hypothesis ($H_0$): The assumption that there is no effect or change.
Alternative Hypothesis ($H_1$): The claim we are testing for.

$$\begin{aligned} H_0 &: p = \frac{1}{2} \\ H_1 &: p < \frac{1}{2} \end{aligned} \quad \quad X \sim B\left(10, \frac{1}{2}\right) \quad \text{where } X \text{ represents the no. Heads thrown.}$$

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Result & Analysis

• Result: $x = 2$
• Significance Level: 5%

Rule: We will reject $H_0$ if the probability of getting our result is less than our significance level.

$$\begin{array}{cl} P(x \leq 2) = 0.0547 & (4 \text{ d.p}) \\ \uparrow & \\ \text{p-value} & \end{array}$$

Definition: The p-value is the probability of obtaining a result as extreme as yours.

Conclusion:

• $0.0547 > 5\%$, so we do not reject our model.
• $\therefore$ NOT SIGNIFICANT and thus do not reject $H_0$.
• At the 5% level, the evidence is not strong enough to suggest the coin is biased against heads.

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2-Tail Test Example

$$\begin{aligned} H_0 &: p = \frac{1}{2} \\ H_1 &: p \neq \frac{1}{2} \end{aligned} \quad \quad X \sim B\left(10, \frac{1}{2}\right) \quad \text{where } X \text{ represents the no. heads.}$$

Result & Analysis

• Significance Level: 5% (Note: In a 2-tail test, we compare the p-value to $2.5\%$ at each tail).
• Result: 9 heads.

$$\begin{aligned} P(X \geq 9) &= 1 - P(X \leq 8) \\ &= 0.0107 \\ 0.0107 &< 2.5\% \end{aligned}$$

Conclusion:

• $\therefore$ SIGNIFICANT and thus reject $H_0$.
• At the 5% level, the evidence suggests the coin is biased. (It appears biased towards heads).