The normal distribution and hypothesis tests

Why this lesson is optional

Bocconi's current test page describes quantitative reasoning as numerical and logical problems using basic mathematics, and it publishes no topic list. A 2024 sample test on Bocconi's website, which the current page does not link to and which may not reflect today's form, included a Type I error and null-hypothesis item and asked for expected values of random variables. Nothing confirms that inferential statistics appears on the current test. Treat this lesson as extra preparation for that possibility, after the core quantitative reasoning lessons.

Expected value and variance

A random variable $X$ takes values with given probabilities. Its expected value $\mu = E(X)$ is the probability-weighted average of its values, and its variance is the expected squared distance from that mean, $\text{Var}(X) = E\big[(X - \mu)^2\big]$. The standard deviation $\sigma$ is the square root of the variance, in the same units as $X$.

If $X$ is 0, 10 or 20 with probabilities 0.2, 0.5 and 0.3, then $E(X) = 0 + 5 + 6 = 11$ and $\text{Var}(X) = 0.2 \times 121 + 0.5 \times 1 + 0.3 \times 81 = 49$, so $\sigma = 7$.

Linear changes behave predictably:

$$E(aX + b) = aE(X) + b, \qquad \text{Var}(aX + b) = a^2\,\text{Var}(X).$$

Adding $b$ slides every value along without spreading them, so the variance is unchanged; multiplying by $a$ stretches every distance by $|a|$ and every squared distance by $a^2$.

For independent $X$ and $Y$, variances add, for a difference as well as a sum:

$$\text{Var}(X + Y) = \text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y).$$

The reason is that $-Y$ has variance $(-1)^2\,\text{Var}(Y) = \text{Var}(Y)$: subtracting a random quantity adds uncertainty. Standard deviations do not add. Independent variables with standard deviations 5 and 12 have a sum, or a difference, with standard deviation $\sqrt{25 + 144} = 13$, not 17. (Without independence a covariance term enters, which this lesson does not need.)

The normal distribution and z-scores

Many measured quantities are roughly normal: a symmetric, bell-shaped density centred on the mean $\mu$, with its spread set by $\sigma$. Probabilities are areas under the curve. Measuring in standard deviations from the mean turns any normal distribution into the standard normal, with mean 0 and standard deviation 1:

$$z = \frac{x - \mu}{\sigma}.$$

If marks have $\mu = 70$ and $\sigma = 8$, a mark of 86 has $z = \frac{16}{8} = 2$, two standard deviations above the mean; a negative $z$ is below the mean.

The 68–95–99.7 rule

For any normal distribution, approximately:

Within	Share inside	Share in each tail beyond
$\mu \pm \sigma$	68%	16%
$\mu \pm 2\sigma$	95%	2.5%
$\mu \pm 3\sigma$	99.7%	0.15%

The tail column comes from symmetry: if 68% lies within one standard deviation, 32% lies outside, half on each side. For a lopsided interval, add pieces measured from the mean: from $\mu - \sigma$ to $\mu$ is 34%, from $\mu$ to $\mu + 2\sigma$ is 47.5%. The more exact figure for the central 95% is $\pm 1.96\sigma$, the number used in intervals and tests.

Sample means and the standard error

The mean $\bar{x}$ of $n$ independent observations, each with standard deviation $\sigma$, varies less from sample to sample than single observations do. By the variance rules, the sum has variance $n\sigma^2$, and dividing by $n$ multiplies the variance by $\frac{1}{n^2}$, so

$$\text{Var}(\bar{x}) = \frac{\sigma^2}{n}, \qquad \text{SE} = \frac{\sigma}{\sqrt{n}}.$$

This is the standard error of the mean. Because of the square root, quadrupling the sample halves the standard error, and cutting it to a third needs nine times the sample. For large samples $\bar{x}$ is close to normal even when the data are not (the central limit theorem), so the rule above applies to it.

Confidence intervals

When $\sigma$ is known, a 95% confidence interval for the population mean is

$$\bar{x} \pm 1.96\,\frac{\sigma}{\sqrt{n}}.$$

With $n = 100$, $\sigma = 15$ and $\bar{x} = 52$, the standard error is 1.5, the margin is $1.96 \times 1.5 = 2.94$, and the interval runs from 49.06 to 54.94. The 95% describes the method: intervals built this way contain the true mean in about 95% of samples. A higher confidence level (2.58 in place of 1.96 for 99%) or a smaller sample makes the interval wider.

The logic of a hypothesis test

A test starts from a null hypothesis $H_0$, the default claim (say, the mean is 500), and an alternative $H_1$ (the mean is not 500). Before seeing the data you fix a significance level $\alpha$, often 5%. Then you ask how surprising the data would be if $H_0$ were true, measuring the gap in standard errors:

$$z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}.$$

In a two-sided test at 5%, reject $H_0$ when $|z| > 1.96$; at 1%, when $|z| > 2.58$. Equivalently, a two-sided test at 5% rejects $\mu_0$ exactly when $\mu_0$ lies outside the 95% confidence interval.

Decision	$H_0$ true	$H_0$ false
Reject $H_0$	Type I error (probability $\alpha$)	Correct
Do not reject $H_0$	Correct	Type II error

A Type I error is rejecting a null hypothesis that is true, and the significance level is the probability of making one when $H_0$ is true. A Type II error is failing to reject a false one. For a fixed sample size, lowering $\alpha$ makes Type I errors rarer and Type II errors more common; a larger sample can reduce both.

The p-value is the probability, computed on the assumption that $H_0$ is true, of a result at least as extreme as the one observed. If $p < \alpha$, reject $H_0$. A p-value is not the probability that $H_0$ is true, it does not measure how large or important an effect is, and failing to reject $H_0$ does not prove it.