Notes
Hypotheses and p-values
- $H_0$ states equality about a parameter. $H_a$ uses $<$, $>$ or $\ne$ and reflects what we suspect.
- The p-value is the probability of results at least as extreme as observed, assuming $H_0$ is true. Small p-values are evidence against $H_0$. It is not the probability that $H_0$ is true.
- Proportion test statistic: $\displaystyle z = \frac{\hat{p} - p_0}{\sqrt{p_0(1 - p_0)/n}}$. For a two-sided test, double the tail area.
Decisions and errors
- If p-value $\le \alpha$, reject $H_0$ (statistically significant). Otherwise fail to reject $H_0$. Never "accept" $H_0$.
- Type I error: reject a true $H_0$ (probability $\alpha$). Type II error: fail to reject a false $H_0$.
Significance
- Statistical significance does not mean practical importance. Large samples can detect tiny effects, and small samples can miss large ones.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Customers suspect a longer time, so $H_a: \mu > 30$.
A p-value is computed assuming $H_0$ is true.
$\displaystyle \text{SE} = \sqrt{\frac{0.25}{400}} = 0.025$ and $\displaystyle z = \frac{0.05}{0.025} = 2$.
$0.07 \le 0.10$, so we reject $H_0$.
A Type I error rejects a true null hypothesis (no improvement).
A small p-value does not measure the size of an effect.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A school claims that $50\%$ of its students ride the bus. A reporter suspects the true percent is different. In a random sample of 400 students, 220 ride the bus.
(a) State the null and alternative hypotheses.
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$H_0: p = 0.5$ and $H_a: p \ne 0.5$, where $p$ is the proportion of all students at the school who ride the bus.
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(b) Compute the test statistic. Using the fact that the area to the right of $z = 2$ is $0.0228$, find the p-value.
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$\displaystyle \hat{p} = \frac{220}{400} = 0.55$. $\displaystyle \text{SE} = \sqrt{\frac{0.5 \cdot 0.5}{400}} = 0.025$. $\displaystyle z = \frac{0.55 - 0.5}{0.025} = 2$. Two-sided p-value $= 2(0.0228) = 0.0456$.
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(c) Using $\alpha = 0.05$, state your decision and conclusion in context. Which type of error could you have made?
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Since $0.0456 \le 0.05$, reject $H_0$. There is convincing evidence that the proportion of students at the school who ride the bus is not $50\%$. Because $H_0$ was rejected, a Type I error is possible.
Check off the points you earned: