Chapter 11: Hypothesis Testing with Single Samples

Welcome to Chapter 11, where we'll dive into the fascinating world of hypothesis testing! Hypothesis testing allows us to make informed decisions about population parameters based on sample data. Get ready to formulate your own hypotheses and test them using the tools and techniques we'll explore.

11.1 Introduction to Hypothesis Testing

A hypothesis is a statement or claim about a characteristic of one or more populations. A hypothesis test is a procedure, based on sample evidence and probability, used to test a statement or claim about a characteristic of one or more populations. Our goal is to determine whether there is enough evidence to reject the null hypothesis.

11.2 Testing a Hypothesis about a Population Mean ($\mu$)

Let's start by testing hypotheses about a population mean, $\mu$. We'll consider two scenarios: when the population standard deviation ($\sigma$) is known, and when it's unknown.

$\sigma$ Known

When $\sigma$ is known, we use the z-test. The steps are:

  1. State the null hypothesis ($H_0$) and the alternative hypothesis ($H_a$). For example:
    • $H_0: \mu = \mu_0$
    • $H_a: \mu \neq \mu_0$ (two-tailed test), or $H_a: \mu > \mu_0$ (right-tailed test), or $H_a: \mu < \mu_0$ (left-tailed test)
  2. Calculate the test statistic: $$z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$$ where $\bar{x}$ is the sample mean, $\mu_0$ is the hypothesized population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size.
  3. Determine the p-value. The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the statistic obtained from the sample, assuming that the null hypothesis is true.
  4. Make a decision: If the p-value is less than or equal to the significance level ($\alpha$), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

$\sigma$ Unknown

When $\sigma$ is unknown, we use the t-test. The test statistic is: $$t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$$ where $s$ is the sample standard deviation. The degrees of freedom (df) for the t-distribution are $n-1$.

11.4 Testing a Hypothesis about a Population Proportion ($p$)

Now, let's test hypotheses about a population proportion, $p$. The steps are similar to testing a mean, but we use a different test statistic:

  1. State the null and alternative hypotheses.
    • $H_0: p = p_0$
    • $H_a: p \neq p_0$ (two-tailed test), or $H_a: p > p_0$ (right-tailed test), or $H_a: p < p_0$ (left-tailed test)
  2. Calculate the test statistic: $$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$$ where $\hat{p}$ is the sample proportion and $p_0$ is the hypothesized population proportion.
  3. Determine the p-value.
  4. Make a decision: If the p-value is less than or equal to $\alpha$, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

11.6 Practical Significance vs. Statistical Significance

It's crucial to differentiate between statistical significance and practical significance. A result may be statistically significant (i.e., the p-value is less than $\alpha$), but it might not be practically significant if the effect size is small or the result is not meaningful in a real-world context. Always consider the context and the magnitude of the effect when interpreting hypothesis tests.