How to Conduct a One-sample Z-test for Population Mean in Interactive Exchanges

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Understanding how to conduct a one-sample Z-test for a population mean is essential for students and professionals involved in statistical analysis. This article provides a step-by-step guide to performing this test through interactive exchanges, making the process clear and engaging.

What Is a One-sample Z-Test?

A one-sample Z-test is a statistical method used to determine whether the mean of a single population differs significantly from a known or hypothesized population mean. It is particularly useful when the population standard deviation is known and the sample size is large (typically n > 30).

Steps to Conduct a One-sample Z-Test

  • State the hypotheses: Formulate the null hypothesis (H0) and alternative hypothesis (Ha).
  • Gather data: Collect a sample and calculate the sample mean (x̄).
  • Identify the population standard deviation (σ): Ensure σ is known for the Z-test.
  • Calculate the Z-statistic: Use the formula Z = (x̄ – μ0) / (σ / √n).
  • Determine the significance level (α): Common choices are 0.05 or 0.01.
  • Find the critical Z-value: Use Z-tables based on α and the nature of the test (one-tailed or two-tailed).
  • Make a decision: Compare the calculated Z with the critical Z-value to accept or reject H0.

Interactive Example

Suppose a manufacturer claims their lightbulbs last an average of 1500 hours. You take a sample of 40 bulbs and find a sample mean of 1480 hours. The population standard deviation is known to be 50 hours. Test at a 0.05 significance level whether the actual mean differs from 1500 hours.

Step 1: State the hypotheses

Null hypothesis (H0): μ = 1500
Alternative hypothesis (Ha): μ ≠ 1500

Step 2: Calculate the Z-statistic

Using the formula: Z = (x̄ – μ0) / (σ / √n)

Z = (1480 – 1500) / (50 / √40) = (-20) / (50 / 6.324) ≈ -20 / 7.905 ≈ -2.53

Step 3: Find the critical Z-value

For a two-tailed test at α = 0.05, the critical Z-values are approximately ±1.96.

Step 4: Make a decision

Since |−2.53| = 2.53 > 1.96, we reject the null hypothesis. There is sufficient evidence to suggest that the average lifespan of the bulbs differs from 1500 hours.

Conclusion

The one-sample Z-test is a powerful tool for comparing a sample mean to a known population mean when the population standard deviation is known. By following the steps outlined and engaging in interactive examples, students can develop a clear understanding of hypothesis testing in statistics.