IB Mathematics Applications & Interpretation AHL 4.18—T and Z Test, Type I and II Errors Notes

Critical Values and Critical Regions

In hypothesis testing, critical values and critical regions play a crucial role in determining whether to reject or fail to reject the null hypothesis.

Critical Values

1. Critical values are the boundaries of the critical region in a distribution.
2. They are determined by the chosen significance level (α) and the type of test being conducted (one-tailed or two-tailed).

For a normal distribution:

• For a two-tailed test at α = 0.05, the critical z-values are ±1.96
• For a one-tailed test at α = 0.05, the critical z-value is 1.645 for a right-tailed test and -1.645 for a left-tailed test.

Critical Regions

The critical region (or rejection region) is the set of values for the test statistic that leads to rejecting the null hypothesis. It's defined by the critical value(s).

Use of Normal and t-Distributions

Normal Distribution (σ known)

• When the population standard deviation (σ) is known, we use the z-test, which follows a normal distribution. This is valid provided the population is normal or the sample size is large enough for the Central Limit Theorem to apply.

The test statistic is calculated as:

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

Where:

• $\bar{x}$ is the sample mean
• $\mu_0$ is the hypothesized population mean
• σ is the known population standard deviation
• n is the sample size

t-Distribution (σ unknown)

• When σ is unknown, we use the t-test, which follows a t-distribution.
• This applies regardless of sample size, though for large samples (n > 30), the t-distribution closely approximates the normal distribution.

The test statistic is:

$$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$$

Where s is the sample standard deviation.

Paired and Unpaired Samples

Unpaired Samples