Assumptions of the One-Sample Z-Test for Population Means

📚 Quick Study Guide

• 🔢 The One-Sample Z-Test is used to determine whether the mean of a population is equal to a specified value, given a known population standard deviation.
• 📝 The formula for the Z-test statistic is: $Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size.
• 📊 The key assumptions are:
• ✔️ Independence: Data points must be independent of each other.
• 🍎 Normality: The sample mean should be approximately normally distributed (often met with $n \geq 30$ due to the Central Limit Theorem).
• 📐 Known Population Standard Deviation: The population standard deviation ($\sigma$) must be known.

🧪 Practice Quiz

1. Which of the following is a key assumption of the One-Sample Z-Test?
   1. A. The population variance is unknown.
   2. B. The data are dependent.
   3. C. The population standard deviation is known.
   4. D. The sample size is less than 30.
2. What happens if the normality assumption is severely violated in a One-Sample Z-Test?
   1. A. The test results are always valid.
   2. B. The test results may be unreliable.
   3. C. The Z-statistic will always be zero.
   4. D. The sample size must be increased.
3. What does the Central Limit Theorem (CLT) suggest about the sampling distribution of the mean?
   1. A. It is always normally distributed, regardless of sample size.
   2. B. It approaches a normal distribution as the sample size increases.
   3. C. It is only applicable when the population is normally distributed.
   4. D. It is always uniform.
4. In the Z-test formula $Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$, what does $\mu$ represent?
   1. A. Sample standard deviation.
   2. B. Sample mean.
   3. C. Population mean.
   4. D. Population standard deviation.
5. What does it mean for data points to be 'independent' in the context of the Z-test?
   1. A. Each data point influences the others.
   2. B. Data points are related through time.
   3. C. Each data point does not affect the others.
   4. D. All data points are identical.
6. Under what condition is the One-Sample Z-Test most appropriate?
   1. A. When the population standard deviation is estimated from the sample.
   2. B. When dealing with proportions.
   3. C. When the population standard deviation is known.
   4. D. When comparing two sample means.
7. If the sample size is small and the population is not normally distributed, what might be a better alternative to the Z-test?
   1. A. Increase the alpha level.
   2. B. Use a One-Sample t-test.
   3. C. Ignore the assumptions.
   4. D. Use a two-sample Z-test.