1-Sample z-test

Next we need to specify the 'null' hypothesis. In this test we are focusing on the mean, so the null hypothesis is that the population mean is equal to some value which we set below. Note that this is the mean of the population our sample was drawn from — a quantity we cannot observe — and not the mean of the sample itself, which we can simply calculate.

Given \(\mu \), where \(\mu \) is the population mean of the variable being analysed:

• The value for n comes from the number of values in the sample chosen above.
• The observed value (OV) comes from the mean of the sample chosen above.

The first assumption is that our sample is independent and randomly chosen from the population we want to draw a conclusion about. This is what licenses us to treat the sample mean as evidence about the population mean at all.

How do we check? We check by investigating the experimental setup. There is no plot or number on this page that can tell us whether this assumption holds — it is a question about how the data was collected, so we have to look at how the study was run.

In Mr. Han’s case study the assumption is satisfied, and it is worth seeing why. The board handed him 25 papers drawn at random from across the country, so every student who sat the exam was equally likely to be in his pile, and one paper landing in it tells us nothing about which others did. That is exactly what the box model assumes: 25 independent draws from the box.

Contrast this with what would happen if Mr. Han had instead marked the 25 papers from his own class. That is also 25 marks, and the arithmetic on this page would run just the same, but they would be one class taught by one teacher, quite possibly streamed by ability. Nothing in the calculation would warn us; the conclusion would simply not be about the country’s students any more. When you work with real data, you would read the methodology section of the accompanying publication and ask exactly this question of it.

The second assumption is that the sample mean follows a normal distribution — that is, if we kept drawing fresh samples of this size from the box and recorded each sample mean, those means would form a normal curve. This is what lets us read the p-value off a normal curve further down the page. Note that the assumption is about the sample mean, not about the individual values.

There are two separate routes to it, and it is worth knowing which one you are relying on.

How do we check?

Idea 1: Large n

• Recall that the central limit theorem tells us that if we take a sufficiently large number of draws from the box, then the sample means will approximately follow a normal distribution, whatever shape the values in the box have. If confused, please do the exercise at Fundamentals > Box Model Part 2.
• One way to gauge whether the central limit theorem holds or not is to see how large our sample is (this is indicated by the "n" in the box model above).
• Many textbooks will say that you can use the rule of thumb that the central limit theorem will apply if we have 30 or more draws. BEWARE - this is not always true! If the distribution of the values is very skewed, you will need much more than 30 draws!

Idea 2: QQ-plot, Boxplot and Histogram
The other route is the shape of the data itself. The central limit theorem is what rescues us when the values in the box are not normal. If they are already normal, we do not need it at all: the sample mean is then exactly normal for any sample size, even a very small one. So the closer our data looks to normal, the less we are leaning on a large n.

• (QQ-plot) If the sample closely follows the QQ line, it suggests the data is normally distributed, and the assumption is on firm ground regardless of n.
• (Boxplot and Histogram) If the data is symmetric, fewer points are needed for the central limit theorem to do its job. These plots can also be used to indicate whether the data appears to be normally distributed.

Be careful about the direction of this reasoning: these plots show us the shape of the sample, and we are using that as a clue about the shape of the box. A small sample can look non-normal by chance even when the box is normal, and vice versa.

As mentioned previously in "The NULL Hypothesis - Setting up the Box" section, it is assumed that the population standard deviation is known. This is the assumption that makes this a z-test: because \(\sigma\) is treated as a known constant, the test statistic sits on a standard normal curve and there are no degrees of freedom to worry about.

How do we check? Like Assumption 1, this is not something the data can answer. We ask where the value of \(\sigma\) came from. A value established from a large body of prior results satisfies the assumption, and Mr. Han’s case study is a rare example of this genuinely holding: the exam has run for years, so the spread of marks across the country is already known to be 7.5, and Mr. Han is marking papers from that same population. A value computed from the sample we are about to test does not satisfy the assumption — in that case it is violated, and the 1-sample t-test is the appropriate test instead.