Intro to Business Statistics: Hypothesis Testing OneSample ZTest for Mean σ Known

What This Is

A one-sample Z-test for mean (σ known) is a statistical method used to determine if a sample mean is significantly different from a known population mean. This is crucial in business decisions, such as evaluating the effectiveness of a marketing campaign, assessing the quality of a product, or determining if a company's sales performance meets expectations. For instance, a retail chain wants to know if average daily sales exceed $10,000 to justify expanding its operations.

Key Formulas & Symbols

• Z = (x̄ – μ) / (σ/√n) where x̄ = sample mean, μ = population mean, σ = population standard deviation, n = sample size.
• σ: population standard deviation (known)
• x̄: sample mean
• μ: population mean (known)
• n: sample size
• df: degrees of freedom (n - 1)
• α: significance level (default = 0.05)
• p-value: probability of observing the data (or more extreme) if H₀ is true
• H₀: null hypothesis (e.g., μ = 10,000)
• H₁: alternative hypothesis (e.g., μ ≠ 10,000)

Step-by-Step Procedure

1. State hypotheses: Define the null and alternative hypotheses (e.g., H₀: μ = 10,000, H₁: μ ≠ 10,000).
2. Choose test: Select the one-sample Z-test for mean (σ known) since the population standard deviation is known.
3. Compute test statistic: Calculate the Z-score using the formula Z = (x̄ – μ) / (σ/√n).
4. Find p-value or critical value: Determine the p-value associated with the calculated Z-score or find the critical Z-score from a standard normal distribution table.
5. Compare to α: Compare the p-value or critical Z-score to the significance level α (default = 0.05).
6. Conclude: Reject the null hypothesis if the p-value < α or the calculated Z-score exceeds the critical Z-score; otherwise, fail to reject the null hypothesis.

Common Mistakes

• Mistake: Using Z when σ is unknown.
• Correction: Use the t-test for mean (σ unknown) instead, which requires estimating the population standard deviation from the sample data.
• Mistake: Misinterpreting p-value as probability H₀ is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if H₀ is true, not the probability that H₀ is true.
• Mistake: Failing to check assumptions (e.g., normality of data).
• Correction: Verify that the data meet the assumptions of the test, such as normality, before proceeding.

Quick Practice Problems

1. A company claims its average employee salary is $50,000. A sample of 36 employees has a mean salary of $52,000 with a population standard deviation of $5,000. Is this claim supported by the data?

Z = (52,000 - 50,000) / (5,000/√36) = 1.33
p-value ≈ 0.092

The p-value is approximately 0.092, which is greater than α = 0.05. Therefore, we fail to reject the null hypothesis, and the company's claim is supported.

1. A marketing firm wants to know if the average response rate to its ads is greater than 2%. A sample of 100 ads has a mean response rate of 2.5% with a population standard deviation of 1.5%. Is this claim supported by the data?

Z = (2.5 - 2) / (1.5/√100) = 1.67
p-value ≈ 0.047

The p-value is approximately 0.047, which is less than α = 0.05. Therefore, we reject the null hypothesis, and the marketing firm's claim is supported.

Last-Minute Cram Sheet

1. Z-test for mean (σ known): Use when population standard deviation is known.
2. T-test for mean (σ unknown): Use when population standard deviation is unknown.
3. α = 0.05: Default significance level.
4. p-value: Probability of observing the data (or more extreme) if H₀ is true.
5. H₀: Null hypothesis (e.g., μ = 10,000).
6. H₁: Alternative hypothesis (e.g., μ ≠ 10,000).
7. df = n - 1: Degrees of freedom for the t-test.
8. σ: Population standard deviation (known).
9. x̄: Sample mean.
10. ⚠️ p-value is NOT the probability that H₀ is true – it’s the probability of observing the data (or more extreme) if H₀ is true.