Revise Confidence intervals and further hypothesis testing for IB Mathematics: Applications and Interpretation (4.9)

You want the population mean $\mu$ but you only have a sample. The sample mean $\bar{x}$ is your single best guess, a point estimate, but it is almost never exactly $\mu$. A confidence interval upgrades that single guess to an honest range: a lower and an upper bound, centred on $\bar{x}$, together with a confidence level such as 90%, 95% or 99% that reports how reliable the method is. In AI HL you never build the interval by hand; you give the GDC the sample mean, the sample standard deviation and the sample size, choose the confidence level, and read the interval off.

GDC: use the t-interval (interval for a mean, population standard deviation unknown). \n Enter: $\bar{x}$ (sample mean), $s_x$ (sample standard deviation), $n$ (sample size), C-Level (e.g. 0.95). \n Structure: interval $= \bar{x} \pm (\text{margin of error})$, where the margin of error $= t \times \dfrac{s_x}{\sqrt{n}}$ and the GDC supplies the factor $t$.

The interpretation must be worded with care. A 95% confidence interval of $(494, 502)$ grams means 'we are 95% confident that the true mean box weight $\mu$ lies between 494 g and 502 g.' It does NOT say that 95% of individual boxes weigh between those values, nor that a particular box has a 95% chance of doing so; the interval is about the mean of the whole population, not about any single member of it. The honest long-run meaning is that if you repeated the sampling many times and built an interval each time, roughly 95% of those intervals would capture the true $\mu$.

Two levers control the width of a confidence interval, and examiners test whether you know which way each one pushes. The confidence level sets how big a net you cast, and the sample size sets how sharp your estimate is. A third factor, how variable the data is, you usually cannot change, but it matters too.

A hypothesis test is a disciplined way to decide whether a claim about a population is contradicted by sample evidence. You set up two competing hypotheses, assume the 'no effect' one is true, and then ask how surprising your sample would be under that assumption. If it would be very surprising, you take that as evidence against the assumption. The whole procedure is built to keep the risk of a false alarm at a level you choose in advance.

The p-value is the single most misread number in the topic, so pin down its meaning: it is the probability of getting a result as extreme as, or more extreme than, the one observed, ASSUMING the null hypothesis is true. It is not the probability that $H_0$ is true, and not the probability that you have made a mistake. A small p-value simply says 'a sample like this would rarely happen if $H_0$ held', which is why a small p-value counts as evidence against $H_0$.

Use a one-sample $t$-test when you want to test whether a single population mean equals some claimed value $\mu_0$, using one sample, when the population standard deviation is unknown and so is estimated from the sample. The GDC does the work: you enter the claimed mean $\mu_0$, the sample mean, the sample standard deviation, the sample size and the tail, and it returns the p-value. Your job is to frame the hypotheses, pick the tail, and interpret.

One-sample test of a mean against a claimed value $\mu_0$: \n $H_0: \mu = \mu_0$ \n $H_1: \mu \ne \mu_0$ (two-tailed), or $H_1: \mu > \mu_0$ or $H_1: \mu < \mu_0$ (one-tailed). \n GDC inputs: $\mu_0$, $\bar{x}$, $s_x$, $n$, and the chosen tail; output: the $p$-value.

Use a two-sample $t$-test to compare the means of two independent groups, for example two fertilisers, two teaching methods, or a treatment against a placebo. You have two samples, each with its own mean, standard deviation and size, and you want to know whether the underlying population means differ. The populations are assumed to be normally distributed; you do not need to know their standard deviations. Unless the question tells you the two populations have equal variances, choose the unpooled option ('Pooled: No') on the GDC.

Comparing two population means $\mu_1$ and $\mu_2$: \n $H_0: \mu_1 = \mu_2$ \n $H_1: \mu_1 \ne \mu_2$ (two-tailed), or $H_1: \mu_1 > \mu_2$ or $H_1: \mu_1 < \mu_2$ (one-tailed). \n GDC inputs: each sample's $\bar{x}$, $s_x$, $n$; the chosen tail; Pooled: No (default). Output: the $p$-value.

On Paper 2 the marks are analytic: each is tied to a specific line of working. A method mark (M) rewards the correct approach; an accuracy mark (A) rewards a correct result and depends on the method mark it follows; follow-through (FT) means an error made earlier does not have to cost the marks that depend on it, provided the later step is carried out correctly on your own figures. The engine also ignores subsequent working once a correct answer appears, and accepts any equivalent wording. That protection only exists if the reasoning is on the page. Study how each mark below is earned by a specific line.

Confidence intervals and $t$-tests are the working core of inference, and the rest of the statistics course leans on the same habits. The standard error $s_x/\sqrt{n}$ that sets an interval's width is the same quantity that drives every test statistic you meet; the p-value-versus-$\alpha$ decision rule reappears unchanged in the chi-squared tests for goodness of fit and independence; and the discipline of stating hypotheses, choosing a tail, reading the GDC and concluding in context is exactly what regression significance and further HL inference reward. Master the two moves here, estimate a mean with an honest interval, and test a claim with an honest p-value, and the inference that follows is variation on a routine you already own.