Chapter 11: Hypothesis Testing

Welcome back to math class! Today, we're diving into Chapter 11, focusing on hypothesis testing for population means and proportions. Remember, hypothesis testing is a powerful tool that allows us to make inferences about a population based on sample data.

Hypothesis Testing for Population Means

Let's start with hypothesis testing for population means. The fundamental steps include:

  1. State the null and alternative hypotheses. The null hypothesis ($H_0$) represents the status quo, while the alternative hypothesis ($H_a$) is what we're trying to find evidence for. For example:
    • $H_0: \mu = 1190$ (The mean sodium content is 1190 mg)
    • $H_a: \mu \neq 1190$ (The mean sodium content is not 1190 mg)
  2. Choose a significance level ($\alpha$). This determines the probability of rejecting the null hypothesis when it's actually true. Common values for $\alpha$ are 0.05 and 0.01.
  3. Calculate the test statistic. This measures how far our sample data deviates from what's expected under the null hypothesis. If we know the population standard deviation we use the Z test, and if we don't, then we use the T test.
  4. Determine the p-value. The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.
  5. Make a decision. If the p-value is less than $\alpha$, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Example: Sodium Content in Oriental Spice Sauce

The nutrition label claims 1190 mg of sodium per package. The FDA samples 200 packages and finds a sample mean of 1167.34 mg and a sample standard deviation of 252.94 mg. We want to test if the label is accurate using $\alpha = 0.01$.

The test statistic is calculated as follows:

$$t = \frac{\bar{x} - \mu}{s / \sqrt{n}} = \frac{1167.34 - 1190}{252.94 / \sqrt{200}} = -1.27$$

With $\alpha = 0.01$, our critical value (t) is 2.576. Since our test statistic (-1.27) is not more extreme than the critical value we fail to reject the null hypothesis. We therefore do not have sufficient evidence to say that the label is inaccurate.

Hypothesis Testing for Population Proportions

Now, let's shift our focus to hypothesis testing for population proportions. This is used when we're dealing with categorical data and want to make inferences about the proportion of a population that possesses a certain characteristic.

The test statistic for proportions is given by:

$$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$$

Where:

  • $\hat{p}$ is the sample proportion
  • $p_0$ is the population proportion (the value used in the null hypothesis)
  • $n$ is the sample size

Example: Commuting Distance

A company is deciding whether to relocate based on concerns that more than 50% of employees will have longer commutes. In a random sample of 398 employees, 201 say their commute would be longer. The significance level is 0.01.

Here:

  • $H_0: p = 0.50$
  • $H_a: p > 0.50$
  • $\alpha = 0.01$

The sample proportion is $\hat{p} = \frac{201}{398} = 0.505$ and so the test statistic is

$$z = \frac{0.505 - 0.50}{\sqrt{\frac{0.50(1-0.50)}{398}}} = 0.199$$

With $\alpha = 0.01$, our critical value (z) is 2.326. Since our test statistic (0.199) is not more extreme than the critical value we fail to reject the null hypothesis. We therefore do not have sufficient evidence to say that more than 50% of employees will have a longer commute.

Keep practicing these concepts, and you'll master hypothesis testing in no time! Remember, statistics is a skill that improves with practice.