Welcome to Section 9-4!
In this section, we're diving into the One-Mean z-Test, a powerful method used to perform hypothesis tests for a population mean, denoted by $\mu$. Let's break down the key steps and considerations:
Assumptions for the One-Mean z-Test
Before you start, ensure these assumptions are met:
- Simple Random Sample: Your data must come from a simple random sample.
- Normality or Large Sample: The population should be normally distributed, or your sample size should be large (typically, $n \geq 30$).
- Known Population Standard Deviation: The population standard deviation, $\sigma$, must be known.
Steps for Conducting a One-Mean z-Test
- State the Null and Alternative Hypotheses:
- Null Hypothesis ($H_0$): This is a statement of no effect or no difference. It usually takes the form $H_0: \mu = \mu_0$, where $\mu_0$ is a specific value.
- Alternative Hypothesis ($H_a$): This is what you're trying to find evidence for. It can take one of three forms:
- Two-Tailed: $H_a: \mu \neq \mu_0$ (the mean is different from $\mu_0$)
- Left-Tailed: $H_a: \mu < \mu_0$ (the mean is less than $\mu_0$)
- Right-Tailed: $H_a: \mu > \mu_0$ (the mean is greater than $\mu_0$)
- Determine the Significance Level ($\alpha$): This is the probability of rejecting the null hypothesis when it's actually true. Common values are 0.05 (5%) and 0.01 (1%).
- Compute the Test Statistic: The z-test statistic is calculated as follows: $$z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$$ where $\bar{x}$ is the sample mean, $\mu_0$ is the value from the null hypothesis, $\sigma$ is the population standard deviation, and $n$ is the sample size. Let's call the calculated z-value $z_0$.
- Determine the Critical Value(s) or P-Value:
- Critical-Value Approach: Find the critical value(s) from the standard normal distribution (z-table) corresponding to your chosen significance level ($\alpha$) and the type of test (one-tailed or two-tailed). The critical values will be $\pm z_{\alpha/2}$ for a two-tailed test, $-z_{\alpha}$ for a left-tailed test, and $z_{\alpha}$ for a right-tailed test.
- P-Value Approach: Find the p-value, which is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated ($z_0$), assuming the null hypothesis is true.
- Make a Decision:
- Critical-Value Approach: If the test statistic ($z_0$) falls in the rejection region (i.e., is more extreme than the critical value(s)), reject the null hypothesis.
- P-Value Approach: If the p-value is less than or equal to the significance level ($\alpha$), reject the null hypothesis.
- Interpret the Results: State your conclusion in the context of the problem. For example, "At the $\alpha$ significance level, there is sufficient evidence to conclude that..."
When to Use the One-Mean z-Test
- For small samples (n < 15), use the z-test *only* if the variable under consideration is normally distributed (or very close to being so).
- For moderate samples (between 15 and 30), the z-test can be used unless the data contains outliers or the variable is far from being normally distributed.
- For large samples (n ≥ 30), the z-test can be used essentially without restriction. However, always check for outliers. If present, analyze the data with and without outliers to assess their impact.
Keep practicing, and you'll master the One-Mean z-Test in no time! Good luck, and see you in the next section!