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
- Critical values are the boundaries of the critical region in a distribution.
- 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 (positive for right-tailed, negative for left-tailed)
If we're testing whether a new teaching method improves test scores with α = 0.05, and we're using a right-tailed test, our critical z-value would be 1.645.
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).
The area of the critical region is equal to the significance level (α).
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 regardless of sample size.
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.
Remember: Use z-test when σ is known, t-test when σ is unknown, regardless of sample size!
When to Use a t-Test vs. a z-Test
The choice between a t-test and a z-test depends on the information available and the sample size:
- Use a z-test when the population standard deviation (σ) is known and the sample size is large (n > 30). This follows the normal distribution.
- Use a t-test when the population standard deviation is unknown and the sample size is small (n ≤ 30). The t-distribution accounts for additional uncertainty in estimating the standard deviation from a small sample.
- If a researcher wants to compare the mean height of a large group of people (σ known), a z-test is appropriate.
- If a psychologist wants to analyze a small experimental group’s improvement in memory (σ unknown), a t-test should be used.
Paired and Unpaired Samples
Unpaired Samples
- Unpaired samples are independent groups being compared.
- For example, comparing test scores of two different classes.
Paired Samples
- Paired samples involve related observations, often before and after measurements on the same subjects.
- For example, measuring blood pressure before and after a treatment.
Matched pairs are treated as a single sample technique. We analyze the differences between the paired observations.
In a study on a weight loss program, we measure participants' weights before and after the program. We'd use a paired t-test to analyze the differences in weight.
Test for Proportion Using Binomial Distribution
- When testing a population proportion, we use the binomial distribution. T
- his is appropriate for situations with binary outcomes (success/failure).
The test statistic is:
$$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$$
Where:
- $\hat{p}$ is the sample proportion
- $p_0$ is the hypothesized population proportion
- n is the sample size
A company claims 80% of customers are satisfied. In a sample of 100 customers, 75 are satisfied. We can use a binomial test to check if this differs significantly from the claim.
Test for Population Mean Using Poisson Distribution
- The Poisson distribution is used for count data, especially for rare events.
- When testing a population mean (λ) for Poisson-distributed data, we use the following test statistic:
$$z = \frac{\bar{x} - \lambda_0}{\sqrt{\lambda_0/n}}$$
Where:
- $\bar{x}$ is the sample mean
- $\lambda_0$ is the hypothesized population mean
- n is the sample size
A factory produces an average of 2 defective items per day. After implementing a new quality control system, they observe 15 defects over 10 days. We can use a Poisson test to see if this is significantly different from the previous average.
Testing Correlation Coefficient
- To test the hypothesis that the population product moment correlation coefficient (ρ) is 0 for bivariate normal distributions, we use technology to calculate the test statistic:
$$t = r \sqrt{\frac{n-2}{1-r^2}}$$
Where:
- $r$ is the sample correlation coefficient
- $n$ is the sample size
This follows a t-distribution with n-2 degrees of freedom.
This test helps determine if there's a significant linear relationship between two variables in the population.
Type I and Type II Errors
Type I Error
- A Type I error occurs when we reject the null hypothesis when it is actually true.
- The probability of a Type I error is equal to the significance level (α).
Type II Error
- A Type II error occurs when we fail to reject the null hypothesis when it is actually false.
- The probability of a Type II error is denoted by β.
Calculating Error Probabilities
For normal distribution with known variance:
$$P(\text{Type I error}) = α (\text{set by the researcher})$$
$$P(\text{Type II error}) = β = P(Z < z_α - \frac{\mu_1 - \mu_0}{\sigma/\sqrt{n}})$$
Where:
- $z_α$ is the critical value
- $\mu_1$ is the true population mean
- $\mu_0$ is the hypothesized mean
Students often confuse Type I and Type II errors. Remember: Type I is rejecting a true null, Type II is failing to reject a false null.
Discrete Random Variables and One-Tailed Tests
For discrete random variables (like Poisson and binomial), hypothesis tests and critical regions are only required for one-tailed tests in the IB curriculum.
The critical region should maximize the probability of a Type I error while keeping it less than the stated significance level.
In a binomial test for $p< 0.5$ with $n = 10$ and $α = 0.05$, the critical region would be $X ≤ 2$, as $P(X ≤ 2) = 0.0547$, which is the largest probability not exceeding $0.05$.
Assumptions for Hypothesis Tests
Before conducting a hypothesis test, certain assumptions must be met:
z-Test Assumptions
- The population follows a normal distribution (or the sample size is large enough for the Central Limit Theorem to apply).
- The population standard deviation (σ) is known.
- The data points are independent.
t-Test Assumptions
- The population follows a normal distribution (especially important for small samples).
- The population standard deviation (σ) is unknown, so we estimate it using the sample standard deviation (s).
- The data points are independent.
