Introduction to Hypothesis Testing: 17 MCQs with answers

It 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 $\mu = \mu_0$.

$H_a: \mu < \mu_0$

The suspicion is a decrease, so the alternative is one-sided in the lower direction: $\mu < \mu_0$.

The 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.

Reject $H_0$ because $p < \alpha$.

Since $0.03 < 0.05$, the p-value is below the significance level, so we reject $H_0$.

Rejecting a true $H_0$

A Type I error is rejecting a true null hypothesis; its probability is $\alpha$.

True

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.

$z = -2.0$

$$z = \frac{490 - 500}{30/\sqrt{36}} = \frac{-10}{5} = -2.0.$$

$\beta$ increases.

For fixed $n$, lowering $\alpha$ reduces Type I errors but raises the Type II error probability $\beta$.

$0.025$ in each tail

A two-tailed test splits $\alpha$ across both tails, so each tail gets $\alpha/2 = 0.025$.

It 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.

There is insufficient evidence to reject $H_0$.

Failing to reject $H_0$ means insufficient evidence, not that $H_0$ is proven true.

It 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.

Reject $H_0$ 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.

The 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.

The 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 $H_0$.

Increase the sample size.

For fixed $n$, $\alpha$ and $\beta$ trade off; increasing $n$ is the way to reduce both.

Reject $H_0$ 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.