AHL 4.18—T and Z test, type I and II errors

When we conduct a hypothesis test, we need a principled way to decide when our evidence is strong enough to reject the null hypothesis $H_0$. This is where critical values and critical regions come in.

Critical Value: A critical value is a threshold value that marks the boundary of the critical region. It is determined by the chosen significance level $\alpha$ and whether the test is one-tailed or two-tailed.

Critical Region: The critical region (also called the rejection region) is the set of values of the test statistic for which we reject $H_0$. The total area of the critical region equals the significance level $\alpha$.

For a normal distribution, the standard critical z-values are:

The most fundamental decision in a one-sample mean test is whether to use a z-test or a t-test. This depends entirely on whether the population standard deviation $\sigma$ is known.

When $\sigma$ is known , use the z-test:

The test statistic is:
$z=\frac{\bar{x}-{\mu }_0}{\sigma /\sqrt{n}}$

Where $\bar{x}$ is the sample mean, ${\mu }_0$ is the hypothesised population mean, $\sigma$ is the known population standard deviation, and $n$ is the sample size. This statistic follows a standard normal distribution $Z\sim N(0,1)$.

When $\sigma$ is unknown , use the t-test:

The test statistic is:
$t=\frac{\bar{x}-{\mu }_0}{s/\sqrt{n}}$

Where $s$ is the sample standard deviation (used to estimate $\sigma$). This statistic follows a t-distribution with $n-1$ degrees of freedom.

Assumptions for the z-test:

• Population follows a normal distribution (or $n$ is large enough for the Central Limit Theorem to apply)
• $\sigma$ is known
• Data points are independent

Assumptions for the t-test:

• Population follows a normal distribution (especially critical for small samples)
• $\sigma$ is unknown
• Data points are independent