Chapter 9: Fundamentals of Hypothesis Testing – One-Sample Tests

Chapter 9: Fundamentals of Hypothesis Testing – One-Sample Tests

Setting Up Hypotheses for the Mean of One Population

Hypothesis testing for the mean of a single population is a fundamental statistical procedure used to make inferences about the population mean based on sample data. The process begins with formulating two competing hypotheses:

• Null Hypothesis (H0): A statement that there is no effect or no difference; typically, it asserts that the population mean equals a specific value.

• Alternative Hypothesis (Ha): A statement that contradicts the null hypothesis; it suggests that the population mean differs from the specified value.

Types of alternative hypotheses:

• Two-tailed test: $H_a: \mu \neq \mu_0$

• Lower-tailed test: $H_a: \mu < \mu_0$

• Upper-tailed test: $H_a: \mu > \mu_0$

Example: Suppose a company claims the average lifetime of its light bulbs is 1000 hours. To test this claim, set $H_0: \mu = 1000$ and $H_a: \mu \neq 1000$ (two-tailed test).

Critical Value Approach and p-Value Approach

There are two main methods for conducting hypothesis tests: the critical value approach and the p-value approach.

• Critical Value Approach:

  1. Specify the significance level $\alpha$ (commonly 0.05).

  2. Determine the appropriate test statistic (z or t) based on whether the population standard deviation $\sigma$ is known.

  3. Calculate the test statistic using sample data.

  4. Find the critical value(s) from statistical tables.

  5. Compare the test statistic to the critical value(s) to decide whether to reject $H_0$.

• p-Value Approach:

  1. Calculate the test statistic as above.

  2. Find the p-value, which is the probability of observing a test statistic as extreme as, or more extreme than, the value observed, assuming $H_0$ is true.

  3. Compare the p-value to $\alpha$:

     • If p-value $\leq \alpha$, reject $H_0$.

     • If p-value $> \alpha$, fail to reject $H_0$.

Formulas:

• If $\sigma$ is known (z-test): $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$

• If $\sigma$ is unknown (t-test): $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

Example: A sample of 36 bulbs has a mean lifetime of 980 hours, $\sigma = 120$ hours. Test $H_0: \mu = 1000$ vs. $H_a: \mu \neq 1000$ at $\alpha = 0.05$.

Using Raw Data to Test Hypotheses for the Mean

When raw data is available, calculate the sample mean ($\bar{x}$) and sample standard deviation ($s$) to use in the test statistic formulas above. The steps remain the same as outlined in the critical value and p-value approaches.

• Compute $\bar{x}$ and $s$ from the data.

• Substitute into the appropriate test statistic formula.

Example: Given raw data for 10 observations, calculate $\bar{x}$ and $s$, then perform a t-test if $\sigma$ is unknown.

Interpreting the Meaning of p-Value

The p-value is the probability, under the null hypothesis, of obtaining a result equal to or more extreme than what was actually observed. It quantifies the evidence against $H_0$:

• Small p-value (typically $\leq 0.05$): Strong evidence against $H_0$; reject $H_0$.

• Large p-value: Insufficient evidence to reject $H_0$.

Example: A p-value of 0.03 means there is a 3% chance of observing such data if $H_0$ is true.

Setting Up and Testing Hypotheses for the Proportion of One Population

Hypothesis testing for a population proportion follows similar steps as for the mean. The hypotheses are:

• Null Hypothesis: $H_0: p = p_0$

• Alternative Hypotheses:

  • Two-tailed: $H_a: p \neq p_0$

  • Lower-tailed: $H_a: p < p_0$

  • Upper-tailed: $H_a: p > p_0$

Test Statistic for Proportion:

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

Where $\hat{p}$ is the sample proportion, $p_0$ is the hypothesized population proportion, and $n$ is the sample size.

Example: In a sample of 200 voters, 120 support a candidate. Test $H_0: p = 0.5$ vs. $H_a: p > 0.5$ at $\alpha = 0.05$.

Additional info: In practice, always check assumptions (e.g., normality for small samples, random sampling) before applying these tests.