📚 Topic Summary
The One-Sample Z-Test is a statistical test used to determine whether there is a significant difference between the mean of a single sample and a known or hypothesized population mean. This test is appropriate when the population standard deviation is known, or when you have a large sample size (typically n > 30) allowing you to estimate it accurately. Essentially, we are checking if our sample data provides enough evidence to reject the null hypothesis – that the sample mean is equal to the population mean.
Understanding the Z-Test involves knowing key vocabulary and performing calculations to obtain a Z-score. This score is then compared to a critical value or used to determine a p-value, which informs our decision on whether to reject the null hypothesis. The following exercises will help you solidify these concepts.
🧠 Part A: Vocabulary
Match the terms with their correct definitions:
| Term |
Definition |
| 1. Null Hypothesis |
A. The probability of obtaining test results at least as extreme as the results actually observed during the test, assuming that the null hypothesis is correct. |
| 2. Alternative Hypothesis |
B. A value used as a cut-off for deciding whether or not to reject the null hypothesis. |
| 3. P-value |
C. The hypothesis that there is no significant difference between specified populations, any observed difference being due to sampling or experimental error. |
| 4. Z-score |
D. The hypothesis that tries to contradict the null hypothesis. |
| 5. Critical Value |
E. A measure of how many standard deviations below or above the population mean a raw score is. |
(Answers: 1-C, 2-D, 3-A, 4-E, 5-B)
✍️ Part B: Fill in the Blanks
Complete the following paragraph using the words provided: mean, standard deviation, sample, population, significant, hypothesis.
The One-Sample Z-Test compares the _____ mean to a known or hypothesized _____ mean. It requires knowledge of the _____ _____ of the population, or a sufficiently large _____ size. The goal is to determine if the difference between the two means is statistically _____, allowing us to test our initial _____.
(Answers: sample, population, standard deviation, sample, significant, hypothesis)
🤔 Part C: Critical Thinking
Explain in your own words why it's important to check the assumptions of the Z-test (e.g., normality, known population standard deviation) before applying it. What could happen if these assumptions are violated?