IB Mathematics Applications & Interpretation AHL 4.18—T and Z Test, Type I and II Errors Notes

Every Hypothesis Test Is The Same Five-Step Argument From Sample To Population

1. A hypothesis test answers one kind of question: does a sample give enough evidence to believe something about the whole population it came from?
2. You never see the population parameter directly, so you start by assuming a specific value for it and ask how surprising your sample would be if that assumption were true.
3. If the sample is surprising enough, you reject the assumption; if not, you keep it for lack of evidence against it.

State The Hypotheses In Population Parameters, Then Choose A Tail

1. The null hypothesis $H_0$ states the value the parameter is assumed to take, written as an equation such as $\mu = \mu_0$, $p = p_0$, $m = m_0$ or $\rho = 0$.
2. The alternative hypothesis $H_1$ states what you are trying to find evidence for, and it decides the number of tails.
   1. A two-tailed test ($\mu \neq \mu_0$) looks for a change in either direction, so the significance level is split between the two ends of the distribution.
   2. A one-tailed test ($\mu > \mu_0$ or $\mu < \mu_0$) looks for a change in one stated direction, so the whole significance level sits in one end.
3. Always write hypotheses in the population parameter, never in the sample statistic: use $\rho$ (not the sample $r$) and $\mu_0$ (not $\bar{x}$), because the test is a claim about the population.
4. The significance level $\alpha$, usually $0.05$ or $0.01$, is the threshold of surprise you fix before seeing the data.

The Decision Rule: Compare The P-Value To The Significance Level

1. The $p$-value is the probability of a sample at least as extreme as yours, computed assuming $H_0$ is true.
2. A small $p$-value means your sample would be very unlikely if $H_0$ held, which is evidence against $H_0$.
3. Never write "accept $H_0$": a test can only fail to find evidence against $H_0$, it can never prove $H_0$ true.
4. Always finish with a sentence in the context of the question, not just the word "reject".

Critical Values And Critical Regions

1. A critical region is the set of test-statistic values extreme enough to reject $H_0$, and a critical value is its boundary.
2. Comparing the $p$-value to $\alpha$ and checking whether the statistic falls in the critical region are two views of the same decision.
3. For a standardised normal statistic, a two-tailed test at $\alpha = 0.05$ has critical values $\pm 1.96$, so a statistic beyond $\pm 1.96$ rejects $H_0$.
4. A one-tailed normal test at $\alpha = 0.05$ has a single critical value of $1.645$ (right tail) or $-1.645$ (left tail).
5. The area of the critical region equals the significance level, so $$P(\text{statistic in critical region} \mid H_0 \text{ true}) = \alpha.$$
6. For discrete distributions the region cannot match $\alpha$ exactly, a case handled later in its own section.

Testing A Single Mean When The Population Deviation Is Known Or Unknown

1. Testing a mean asks whether a sample mean $\bar{x}$ is far enough from the claimed $\mu_0$ to reject $H_0$.
2. The whole decision turns on one fact: do you know the true population standard deviation $\sigma$, or must you estimate it from the sample?

The Known-Deviation Case: A Z-Test

1. When $\sigma$ is genuinely known, the standardised distance from the claim to the sample mean follows a normal distribution.
2. The test statistic measures that distance in standard errors: $$z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}},$$
   1. where the denominator $\sigma / \sqrt{n}$ is the standard error of the mean, which comes directly from the central limit theorem result $\bar{X} \sim N\!\left(\mu, \dfrac{\sigma^2}{n}\right)$.
3. A $z$-test is valid when the population is normal, or when the sample is large enough for the central limit theorem to make $\bar{X}$ approximately normal.

The Unknown-Deviation Case: A T-Test

1. In almost every real study $\sigma$ is unknown, so you replace it with the unbiased sample estimate $s_{n-1}$.
2. This gives the $t$-statistic $$t = \frac{\bar{x} - \mu_0}{s_{n-1} / \sqrt{n}},$$
   1. which follows a $t$-distribution with $n - 1$ degrees of freedom rather than a normal distribution.
3. Use a $t$-test whenever $\sigma$ is unknown, regardless of the sample size; the sample size does not decide between $z$ and $t$.
4. The extra spread of the $t$-distribution accounts for the added uncertainty of estimating $\sigma$, and it approaches the normal curve as $n$ grows.

Comparing Two Means: Unpaired Versus Paired Samples

Many tests compare two group means rather than one mean against a fixed value, and the design of the data decides which test to run.

Unpaired Samples: Two Independent Groups

1. Unpaired (independent) samples come from two separate groups with no natural pairing, such as the exam marks of two different classes.
2. The hypotheses compare the two population means, with $H_0\!: \mu_1 = \mu_2$ against a one- or two-tailed alternative.
3. A two-sample $t$-test assumes the two populations have the same variance, an assumption you must state as "equal", not merely "similar".
4. The calculator returns the pooled $t$-statistic and $p$-value once you enter the two samples or their summary statistics.

Paired Samples: One Sample Of Differences

1. Paired samples are two measurements on the same subjects, such as each patient's blood pressure before and after treatment.
2. The trick is to reduce each pair to a single difference $d = x_{\text{before}} - x_{\text{after}}$, turning matched pairs into one sample of differences.
3. You then run a one-sample $t$-test on those differences, with hypotheses in the population mean difference: $H_0\!: \mu_d = 0$ against the alternative that fits the claim.
4. Pairing removes the variation between subjects, so a paired test can detect a change that an unpaired test would miss.

Testing A Proportion With The Binomial Distribution

1. A proportion test asks whether a population proportion $p$ differs from a claimed value $p_0$, using count data of successes and failures.
2. The hypotheses are stated in the proportion, $H_0\!: p = p_0$ against a one-tailed alternative, and in AI the proportion test is always one-tailed.
3. Under $H_0$ the number of successes follows a binomial distribution $X \sim B(n, p_0)$, so the $p$-value is an exact binomial tail probability.
4. For a lower-tail alternative the $p$-value is $P(X \leq x)$ using the observed count $x$; for an upper-tail alternative it is $P(X \geq x)$.
5. You compute this tail directly with the calculator's binomial cumulative function rather than any by-hand formula.

Testing A Poisson Mean, One Tail Only

1. A Poisson test asks whether the mean rate $m$ of events in a fixed interval differs from a claimed $m_0$, using a single count of events.
2. The hypotheses are stated in the Poisson mean, $H_0\!: m = m_0$, and like the binomial test it is one-tailed only in AI.
3. Under $H_0$ the count follows $X \sim \text{Po}(m_0)$, so the $p$-value is an exact Poisson tail probability such as $P(X \geq x)$ for an "increased rate" claim.
4. Use the IB symbol $m$ for the mean, not $\lambda$, and rescale $m_0$ if the observation covers a different interval length from the quoted rate.
5. For a large mean you may instead use the normal approximation $X \approx N(m_0, m_0)$, since a Poisson has equal mean and variance, and run a $z$-test on it.

Testing A Correlation For Bivariate Normal Data

1. A correlation test asks whether two variables that follow a bivariate normal distribution are linearly related in the population.
2. The hypotheses are stated in the population correlation coefficient $\rho$: $H_0\!: \rho = 0$ against a one- or two-tailed alternative.
3. Writing the hypotheses in the sample $r$ instead of the population $\rho$ is a marked error, because the test is a claim about the population.
4. The underlying statistic is $t = r\sqrt{\dfrac{n - 2}{1 - r^2}}$ with $n - 2$ degrees of freedom, but it is not in the booklet and the GDC returns the $p$-value for you.
5. A significant result means evidence of a linear relationship, so interpret it as correlation and never as proof of causation.

Critical Regions For Discrete Distributions

1. For a discrete distribution such as the binomial or Poisson, the critical region is a set of whole-number counts, so its probability cannot equal $\alpha$ exactly.
2. The rule is to make the region as large as possible while keeping its probability at most $\alpha$: $$P(X \in \text{critical region} \mid H_0) \leq \alpha.$$
3. For an upper-tail test you find the smallest critical value $c$ such that $P(X \geq c \mid H_0) \leq \alpha$.
4. Any count landing in that region leads you to reject $H_0$, which is exactly how the discrete tests above make their decision.

Type I And Type II Errors

1. Because a test decides from a sample, it can reach the wrong conclusion in two distinct ways.
2. A Type I error is rejecting $H_0$ when $H_0$ is actually true, so it is a false alarm.
3. A Type II error is failing to reject $H_0$ when $H_0$ is actually false, so it is a missed effect.
4. The probability of a Type I error equals the significance level, $P(\text{Type I error}) = \alpha$, which is why choosing $\alpha$ controls the false-alarm rate.
5. The probability of a Type II error is written $\beta$, and it can only be found when a specific true value of the parameter is given.

Calculating A Type I Error Probability

1. For a discrete test with a fixed critical region, the Type I error probability is the probability of landing in that region computed under $H_0$.
2. So you evaluate a tail probability using the null parameter value, exactly as $P(X \geq 15 \mid p = 0.5)$ was found above.

Calculating A Type II Error Probability

1. A Type II error means the statistic misses the critical region even though $H_0$ is false, so you compute the complementary probability under the true value.
2. You therefore switch the distribution to the given true parameter and find the probability of not being in the critical region.

Choosing The Right Test

1. The whole subtopic reduces to matching the situation to a parameter and a distribution, then running the pipeline.
2. Read the context for what is being measured, what parameter the claim concerns, and whether $\sigma$ is known, then pick the test.