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Lesson 1 of 7 · 13 min

The six steps and stating the hypotheses

A hypothesis test asks whether a sample result lies so far from a claimed population value that chance alone is a poor explanation.

In short

A hypothesis is a claim about a population parameter (a mean, a variance). We judge it with a sample statistic.
The null hypothesis $H_0$ is presumed true until the data argue strongly against it. It always holds the equality: =, ≤ or ≥.
The alternative hypothesis $H_a$ holds the suspected or hoped-for condition. Together the two cover every possible value of the parameter.
Six steps: state the hypotheses → identify the test statistic and its distribution → specify the significance level → state the decision rule → collect data and calculate → decide.
The only outcomes are reject $H_0$ or fail to reject $H_0$ . A test never proves the null.

What a hypothesis test is for

Statistical inference means judging a whole population from a sample. It has two branches. Estimation gives point estimates and confidence intervals (see Estimation and Inference: Confidence intervals). Hypothesis testing checks whether the sample is consistent with one specific claimed value.

A hypothesis is a statement about one or more populations, written in terms of population parameters such as $\mu$ or $\sigma^2$ . We never test a sample value: the sample mean is known exactly, so there is nothing to test about it.

The core question never changes. If the claimed value were true, how surprising would our sample be? A small gap between the sample statistic and the claim is easily explained by sampling luck. A large gap is not, and we then reject the claim.

The conclusion is about probabilities, never certainty. A test can only say whether the evidence is strong enough at a level of risk we chose in advance.

Think of a courtroom. The defendant (the null) is presumed innocent. The prosecutor (the analyst) must bring enough evidence to convict (reject). A not-guilty verdict does not prove innocence; it says the evidence fell short. That is exactly what 'fail to reject' means.

Stating the null and the alternative

The null hypothesis $H_0$ is the claim being tested. It stands until the sample gives convincing evidence against it, and it always contains the equal sign: =, ≤ or ≥.

The alternative hypothesis $H_a$ is what we conclude if the null is rejected. Analysts put the condition they suspect or hope to show in the alternative, because rejecting the null is a much stronger statement than failing to reject it.

The two must be mutually exclusive (no value belongs to both) and collectively exhaustive (every possible value belongs to one of them). So if $H_0: \mu \leq 5\%$ , the alternative must be $H_a: \mu > 5\%$ .

The wording of the question sets the form. 'Different from' gives a two-sided (two-tailed) test. 'Greater than', 'exceeds', 'less than' or 'below' give a one-sided (one-tailed) test. A one-sided test reflects a firmer belief about the direction of the effect.

The three forms of a test of a mean
Question asked	Null	Alternative	Rejection region
Is $\mu$ different from $\mu_0$ ?	$H_0: \mu = \mu_0$	$H_a: \mu \neq \mu_0$	Both tails
Is $\mu$ greater than $\mu_0$ ?	$H_0: \mu \leq \mu_0$	$H_a: \mu > \mu_0$	Right tail only
Is $\mu$ less than $\mu_0$ ?	$H_0: \mu \geq \mu_0$	$H_a: \mu < \mu_0$	Left tail only

For

H_a: \mu > 2\%

(the retailer's claim) the whole 5% rejection region sits in the right tail. The sample's t of 2.20 lands beyond 1.860, so the null is rejected; a large negative t could never reject this null.

The six steps

Every test in this module, and the regression tests later in Quant, follows the same six steps. Learn the order: exam questions ask which step a given task belongs to.

Notice that steps 1–4 are fixed before the data are examined. Setting the rules first stops an analyst from picking a significance level or a tail that happens to give the answer they want.

Step 1: State the hypotheses. Write $H_0$ and $H_a$ in terms of the population parameter.
Step 2: Identify the test statistic and its distribution. Choose the formula that fits what is tested (one mean, two means, a variance…) and note its distribution and degrees of freedom.
Step 3: Specify the significance level $\alpha$ : the risk of rejecting a true null you are willing to take, typically 10%, 5% or 1%.
Step 4: State the decision rule. Turn $\alpha$ into critical values, the cut-offs that separate 'reject' from 'fail to reject'.
Step 5: Collect the data and calculate the test statistic.
Step 6: Make a decision. Compare the statistic with the critical values (or the p-value with $\alpha$ ) and say in plain words what the result means.

Steps 1–4 are fixed before the sample is examined; only steps 5 and 6 use the data. Setting the rules first stops the analyst choosing a level or a tail that suits the answer.

Choosing the test statistic

A test statistic is a number calculated from the sample that, together with the decision rule, decides whether we reject the null. Most share one shape: (sample statistic − hypothesised value) ÷ standard error. It measures how many standard errors the sample result sits from the claim.

Which formula you use depends on what is being tested. The table lists the tests in this module; each gets its own lesson. Two more tests from the same family, for a correlation and for independence in a contingency table, are taught in the next module.

Which test for which question
What is tested	Statistic	Distribution	Degrees of freedom
One mean	$t = \frac{\bar X - \mu_0}{s/\sqrt{n}}$	t	$n - 1$
Difference between two means (independent samples)	Pooled t	t	$n_1 + n_2 - 2$
Mean of differences (paired samples)	$t = \frac{\bar d - \mu_{d0}}{s_d/\sqrt{n}}$	t	$n - 1$
One variance	$\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}$	Chi-square	$n - 1$
Two variances (independent samples)	$F = \frac{s_1^2}{s_2^2}$	F	$n_1 - 1$ and $n_2 - 1$

Formula

General form of a test statistic: $\text{Test statistic} = \dfrac{\text{Sample statistic} - \text{Hypothesised value}}{\text{Standard error of the sample statistic}}$
Measures the distance between the sample result and the claim in standard-error units. The further out, the stronger the evidence against $H_0$ .

Worked examples

Example 1 Writing the hypotheses

A credit analyst suspects that the mean loan-loss rate of Danish regional banks has risen above 1.5%. Which hypotheses should she test, and where is the rejection region?

Example 2 Matching the test to the question

Name the test statistic and degrees of freedom. (a) 30 monthly returns: is the mean return zero? (b) 20 weekly returns: is volatility above 2%? (c) Two unrelated samples of 12 and 18 firms: are mean margins equal? (d) Two independent samples of 25 and 31 returns: are the variances equal?

Example 3 All six steps in one test

A Swedish retailer claims a new strategy lifted mean quarterly sales growth above 2%. Over 9 quarters growth averaged 3.1% with a sample standard deviation of 1.5%. Test the claim at 5% (one-tailed critical t with 8 df = 1.860).

On the exam

Expect questions that turn a sentence into $H_0$ and $H_a$ , ask which step a task belongs to, or ask which statistic and degrees of freedom fit a situation.

Putting the analyst's belief in the null. The null is what you try to reject; the suspected condition goes in $H_a$ .
Writing a null without equality, such as $H_0: \mu > 0$ . The null always holds =, ≤ or ≥.
Saying the null is 'accepted' or 'proved'. The only outcomes are reject or fail to reject.
Placing the calculation in step 2. Step 2 only identifies the statistic; it is calculated in step 5.

Check yourself

Q1An analyst wants to test whether mean excess returns have fallen below 1%. Which hypotheses are most appropriate?
Q2Which statement about the null and alternative hypotheses is most accurate?
Q3Two unrelated samples of 14 and 20 funds are used to test whether their mean returns differ. The degrees of freedom are:
Q4In the six-step process, the critical values are set when:
Q5Sixteen returns have a mean of 2.4% and a standard deviation of 1.6%. The t-statistic for $H_0: \mu = 2\%$ is closest to:
Q6A test fails to reject $H_0: \mu = 0$ . The most accurate conclusion is that:

0/6Check yourself

Answer the 6 questions: +10 XP for each one you get right first time.

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