Kit Library / Mathematics / Statistics & Probability
⚡ Topic Learning KitShared by a Veda learner · Generated with Veda AI
Veda Bites are swipeable micro-lessons — each one teaches exactly one idea. The bites below come from this kit — swipe through them in the app.
Data on trial: innocent until proven guilty.
Hypothesis testing is a formal procedure for deciding whether sample data provides enough evidence to reject a claim about a population.
↳ Hypothesis testing never proves $H_0$ — it only checks whether the data is unlikely under $H_0$.
Two claims walk into a courtroom.
↳ Write $H_0$ and $H_a$ before touching the data — they must be mutually exclusive and exhaustive.
How far is your sample from the claim?
The test statistic converts the gap between your sample and $H_0$ into a standardized number — how many standard errors away the sample sits.
↳ A larger $|z|$ means the sample is further from $H_0$ — stronger evidence against it.
Set your error budget before you test.
↳ $\alpha$ is chosen **before** the test — never after seeing the data.
The probability that kills (or saves) a hypothesis.
Decision rule: if $p \leq \alpha$, reject $H_0$; if $p > \alpha$, fail to reject $H_0$.
↳ A small p-value means the data is unlikely under $H_0$ — but it is NOT the probability that $H_0$ is true.
The most misquoted verdict in statistics.
↳ Absence of evidence is not evidence of absence — failing to reject $H_0$ is not proof that $H_0$ is true.
Two ways to be wrong — pick your poison.
↳ Lowering $\alpha$ reduces Type I errors but increases Type II errors — there is always a trade-off.
Five steps from question to verdict.
↳ Always state the conclusion in the language of the original problem — not just 'reject' or 'fail to reject'.
A factory, a sample, and a verdict.
↳ A $z$-score of 2.0 exceeds the critical value 1.96 at $\alpha = 0.05$ (two-tailed) — evidence is significant.
Which side of the curve are you watching?
↳ Choose the tail direction from $H_a$ **before** looking at the data — never after.
From medicine to your Netflix feed.
↳ Anywhere a claim needs evidence, hypothesis testing is the referee.
If p is low, H₀ must go.
'If the p is low, the null must go. If the p is high, the null will fly.'
↳ p ≤ α → reject $H_0$; p > α → fail to reject $H_0$.
Score full marks — every time.
↳ Examiners award marks for the **reasoning chain**, not just the final verdict.
Significant doesn't always mean important.
↳ Always ask: is the effect size large enough to matter in the real world?
Build the base before the test.
↳ Hypothesis testing is built on sampling distributions — master those first.
Hypothesis testing is a statistical method for deciding whether sample data provides enough evidence to reject a claim about a population. It frames two competing hypotheses — a null and an alternative — and uses a test…
~15 min · full explanation, examples & memory tricks in the app
All 17 questions, laddered from warm-up to topper-level, each with its answer and the reasoning behind it.
Q1. Which statement best describes the null hypothesis $H_0$?
BeginnerIt states 'no effect' or 'no difference' and contains an equality.
$H_0$ is the default claim of no effect or no difference and always contains an equality such as $\mu = \mu_0$.
Q2. A researcher suspects a new drug lowers blood pressure. Which alternative hypothesis fits?
Beginner$H_a: \mu < \mu_0$
The suspicion is a decrease, so the alternative is one-sided in the lower direction: $\mu < \mu_0$.
Q3. What does the significance level $\alpha$ represent?
BeginnerThe threshold probability for rejecting $H_0$, commonly 0.05.
$\alpha$ is the pre-chosen threshold for rejecting $H_0$, often 0.05, and equals the Type I error probability.
Q4. A test gives $p = 0.03$ at $\alpha = 0.05$. What is the decision?
BeginnerReject $H_0$ because $p < \alpha$.
Since $0.03 < 0.05$, the p-value is below the significance level, so we reject $H_0$.
Q5. What is a Type I error?
BeginnerRejecting a true $H_0$
A Type I error is rejecting a true null hypothesis; its probability is $\alpha$.
Q6. The p-value is the probability of observing data at least as extreme as the sample, computed under the assumption that $H_0$ is true — it is not the probability that $H_0$ is true.
BeginnerTrue
The p-value is the probability of observing data at least as extreme as the sample, computed under the assumption that $H_0$ is true — it is not the probability that $H_0$ is true.
Q7. A quality-control engineer claims the mean fill is 500 mL. A sample of 36 bottles gives $\bar{x} = 490$ with $\sigma = 30$. Compute the test statistic for $H_0: \mu = 500$.
Intermediate$z = -2.0$
$$z = \frac{490 - 500}{30/\sqrt{36}} = \frac{-10}{5} = -2.0.$$
Q8. For a fixed sample size, what happens to the Type II error probability $\beta$ when $\alpha$ is decreased?
Intermediate$\beta$ increases.
For fixed $n$, lowering $\alpha$ reduces Type I errors but raises the Type II error probability $\beta$.
Q9. A two-tailed test is run at $\alpha = 0.05$. How much probability is placed in each tail?
Intermediate$0.025$ in each tail
A two-tailed test splits $\alpha$ across both tails, so each tail gets $\alpha/2 = 0.025$.
Q10. Which statement about the p-value is correct?
IntermediateIt is the probability of data at least as extreme as observed, assuming $H_0$ is true.
The p-value is computed under the assumption that $H_0$ is true; it is not the probability that $H_0$ is true.
Q11. A study reports $p = 0.20$ at $\alpha = 0.05$. Which conclusion is correct?
IntermediateThere is insufficient evidence to reject $H_0$.
Failing to reject $H_0$ means insufficient evidence, not that $H_0$ is proven true.
Q12. A company tests whether a new website layout improves conversion. They choose a one-tailed test. What is the main consequence?
IntermediateIt is easier to reject in the chosen direction but blind to effects in the other direction.
A one-tailed test puts all of $\alpha$ in one direction, making rejection easier there but insensitive to the opposite effect.
Q13. A researcher computes $z = -2.0$ for a two-tailed test at $\alpha = 0.05$ with critical values $\pm 1.96$. What is the decision?
AdvancedReject $H_0$ because $|z| = 2.0 > 1.96$.
For a two-tailed test, compare $|z|$ with the critical value: $2.0 > 1.96$, so $H_0$ is rejected.
Q14. A very large sample produces a statistically significant result with a tiny effect size. What is the best interpretation?
AdvancedThe effect is statistically significant but may not be practically important.
Statistical significance answers whether an effect exists; with large $n$, tiny effects can be significant without being practically meaningful.
Q15. A trial fails to reject $H_0$ at $\alpha = 0.05$. Which statement is most accurate?
AdvancedThe sample may have been too small or the test lacked power.
Non-rejection can reflect insufficient evidence, low power, or a small sample — it does not prove $H_0$.
Q16. A team wants to reduce both Type I and Type II error probabilities simultaneously. What should they do?
AdvancedIncrease the sample size.
For fixed $n$, $\alpha$ and $\beta$ trade off; increasing $n$ is the way to reduce both.
Q17. A researcher is testing whether a new teaching method improves student performance. She sets up the hypotheses as follows: $$H_0: \mu = 75$$ (no improvement) and $$H_a: \mu > 75$$ (improvement). After collecting data from 50 students, she calculates a test statistic of $$z = 1.85$$. Given that the significance level is $$\alpha = 0.05$$, what should the researcher conclude?
AdvancedReject $H_0$ because $z$ is greater than the critical value.
The critical value for a one-tailed test at $\alpha = 0.05$ is approximately 1.645. Since the calculated $z = 1.85$ is greater than 1.645, the researcher should reject $H_0$, indicating significant evidence that the new teaching method improves student performance.
20 flashcards in this kit — the app reviews them with spaced repetition so the right card returns on the right day.
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