z-test

A z-test is a statistical method used to determine whether there is a significant difference between the means of two groups or whether a sample mean significantly differs from a known population mean. It is based on the assumption that the sampling distribution of the sample mean is normally distributed, especially when the sample size is large (typically $n \geq 30$).

Key Concepts of the Z-Test

1. Z-Score:

   • The z-score measures how many standard deviations an element is from the mean. It is calculated as:
   $$z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}$$
   • Where:
     • $\bar{x}$: Sample mean
     • $\mu$: Population mean
     • $\sigma$: Population standard deviation
     • $n$: Sample size

2. Assumptions:

   • The data should be normally distributed, or the sample size should be large enough for the Central Limit Theorem to apply.
   • The population standard deviation ($\sigma$) is known.

Types of Z-Tests

1. One-Sample Z-Test:

   • Used to determine whether the mean of a single sample is significantly different from a known population mean.
   • Hypotheses:
     • Null hypothesis ($H_0$): The sample mean is equal to the population mean ($\bar{x} = \mu$).
     • Alternative hypothesis ($H_a$: The sample mean is not equal to the population mean ($\bar{x} \neq \mu$).

2. Two-Sample Z-Test:

   • Used to compare the means of two independent samples.
   • Hypotheses:
     • Null hypothesis ($H_0$): The means of the two samples are equal ($\mu_1 = \mu_2$).
     • Alternative hypothesis ($H_a$): The means of the two samples are not equal ($\mu_1 \neq \mu_2$).
   • The formula for the z-score in a two-sample z-test is:
   $$z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$$
   • Where:
     • $\bar{x}_1, \bar{x}_2$: Sample means
     • $\sigma_1, \sigma_2$: Population standard deviations
     • $n_1, n_2$: Sample sizes

Steps to Conduct a Z-Test

1. State the Hypotheses:

   • Define the null and alternative hypotheses.

2. Collect Data:

   • Gather the necessary sample data.

3. Calculate the Z-Score:

   • Use the appropriate formula based on whether it’s a one-sample or two-sample test.

4. Determine the Critical Value:

   • Use a z-table (standard normal distribution table) to find the critical z-value based on the desired significance level ($\alpha$), typically 0.05 for a 95% confidence level.

5. Make a Decision:

   • Compare the calculated z-score to the critical z-value:
     • If $|z| > z_{\text{critical}}$: Reject the null hypothesis.
     • If $|z| \leq z_{\text{critical}}$: Fail to reject the null hypothesis.

Example of a One-Sample Z-Test

Scenario: A factory claims that the average lifespan of its light bulbs is 1000 hours. A quality control manager tests a sample of 40 bulbs and finds an average lifespan of 970 hours with a population standard deviation of 120 hours. Is there enough evidence to reject the factory's claim at the 0.05 significance level?

1. State the Hypotheses:

   • $H_0: \mu = 1000$ (the average lifespan is 1000 hours)
   • $H_a: \mu \neq 1000$ (the average lifespan is not 1000 hours)

2. Collect Data:

   • Sample mean ($\bar{x} = 970$), population standard deviation ($\sigma = 120$), sample size ($n = 40$).

3. Calculate the Z-Score:

   $$z = \frac{970 - 1000}{120/\sqrt{40}} = \frac{-30}{19} \approx -1.58$$

4. Determine the Critical Value:

   • For a two-tailed test at $\alpha = 0.05$, the critical z-values are approximately ±1.96.

5. Make a Decision:

   • Since $-1.58$ is within the range $[-1.96, 1.96]$, we fail to reject the null hypothesis. There is not enough evidence to conclude that the average lifespan is different from 1000 hours.

Conclusion

The z-test is a powerful statistical tool for hypothesis testing when the population standard deviation is known and sample sizes are sufficiently large. By following the structured steps of formulating hypotheses, calculating z-scores, and making decisions based on critical values, researchers can effectively test claims and make data-driven conclusions.