Chapter 9 Review: Hypothesis Testing for One Sample

Hypothesis Testing for One Sample

Introduction to Hypothesis Testing

Hypothesis testing is a fundamental statistical method used to make inferences about population parameters based on sample data. In one-sample hypothesis testing, we typically assess claims about the population mean or proportion using sample statistics.

• Null Hypothesis (H0): The default assumption that there is no effect or no difference. For example, $H_0: \\mu = \\mu_0$.

• Alternative Hypothesis (Ha): The claim we are testing for, such as $H_a: \\mu \\ eq \\mu_0$, $H_a: \\mu > \\mu_0$, or $H_a: \\mu < \\mu_0$.

• Test Statistic: A value calculated from sample data that is used to decide whether to reject the null hypothesis.

• Significance Level (\\alpha): The probability of rejecting the null hypothesis when it is true, commonly set at 0.05 or 0.01.

Steps in Hypothesis Testing

1. State the hypotheses: Clearly define $H_0$ and $H_a$.

2. Choose the significance level (\\alpha): Decide on the risk of Type I error.

3. Calculate the test statistic: For means, use the z-test or t-test depending on sample size and population standard deviation knowledge.

4. Find the p-value or critical value: The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed value under $H_0$.

5. Make a decision: If p-value $< \\alpha$, reject $H_0$; otherwise, fail to reject $H_0$.

Test Statistics for One Sample Mean

• Z-Test: Used when the population standard deviation ($\\sigma$) is known and the sample size is large ($n \\geq 30$). Formula: $z = \\frac{\\bar{x} - \\mu_0}{\\sigma / \\sqrt{n}}$

• T-Test: Used when $\\sigma$ is unknown and the sample size is small ($n < 30$). Formula: $t = \\frac{\\bar{x} - \\mu_0}{s / \\sqrt{n}}$

Critical Values and Rejection Regions

Critical values are determined by the chosen significance level and the type of test (one-tailed or two-tailed). For example, for $\\alpha = 0.05$ in a two-tailed z-test, the critical values are $\\pm 1.96$.

• One-tailed test: Used when the alternative hypothesis is directional ($H_a: \\mu > \\mu_0$ or $H_a: \\mu < \\mu_0$).

• Two-tailed test: Used when the alternative hypothesis is non-directional ($H_a: \\mu \\ eq \\mu_0$).

Types of Errors

• Type I Error (\\alpha): Rejecting $H_0$ when it is true.

• Type II Error (\\beta): Failing to reject $H_0$ when it is false.

Example: One-Sample Z-Test

• Given: $\\bar{x} = 52$, $\\mu_0 = 50$, $\\sigma = 4$, $n = 25$

• Calculate: $z = \\frac{52 - 50}{4 / \\sqrt{25}} = \\frac{2}{0.8} = 2.5$

• Decision: If $z = 2.5$ and $\\alpha = 0.05$ (two-tailed), since $2.5 > 1.96$, reject $H_0$.

Table: Comparison of Z-Test and T-Test

Test	Population Std. Dev. Known?	Sample Size	Distribution
Z-Test	Yes	Large ($n \\geq 30$)	Standard Normal
T-Test	No	Small ($n < 30$)	Student's t

Interpreting Results

• If the p-value is less than $\\alpha$, the result is statistically significant.

• Always state the conclusion in context of the original claim.

Additional info: The original file appears to be a review worksheet or test with answer key for Chapter 9, focusing on hypothesis testing for one sample, including calculation of test statistics, critical values, and decision rules.