Intro to Business Statistics: Hypothesis Testing Test Statistic and Critical Value Method

What This Is

The Test Statistic and Critical Value Method is a statistical technique used to make decisions about a population based on a sample of data. This method is essential in business decisions, such as determining if a new marketing campaign is effective, if a product meets quality standards, or if a company's sales are above a certain threshold. For example, a retail chain wants to know if average daily sales exceed $10,000 to decide whether to expand their store hours.

Key Formulas & Symbols

• Z = (x̄ – μ) / (σ/√n) where x̄ = sample mean, μ = population mean, σ = population standard deviation, n = sample size.
• t = (x̄ – μ) / (s/√n) where x̄ = sample mean, μ = population mean, s = sample standard deviation, n = sample size.
• χ² = Σ [(xi – x̄)² / s²] where xi = individual data points, x̄ = sample mean, s = sample standard deviation.
• F = (MSB / MSW) where MSB = Mean Square Between, MSW = Mean Square Within.
• p-value = P(TS ≥ |TS|) where TS = Test Statistic.
• α = 0.05 (default significance level).
• df = n - 1 (degrees of freedom for t-distribution).
• Critical Value (CV) = Zα/2 (for Z-distribution) or tα/2, df (for t-distribution).

Step-by-Step Procedure

1. State hypotheses: Clearly define the null (H₀) and alternative (H₁) hypotheses.
2. Choose test: Select the appropriate statistical test (e.g., Z-test, t-test, χ²-test, F-test) based on the data and research question.
3. Compute test statistic: Calculate the test statistic using the chosen formula.
4. Find p-value or critical value: Determine the p-value or critical value using a standard normal distribution (Z-distribution) or a t-distribution table.
5. Compare to α: Compare the p-value or critical value to the significance level (α = 0.05).
6. Conclude: Make a decision based on the comparison (e.g., reject H₀ if p-value < α, fail to reject H₀ if p-value ≥ α).

Common Mistakes

• Mistake: Using Z when σ is unknown.
• Correction: Use t-test instead, which is robust to small sample sizes and unknown population standard deviation.
• 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, equal variances).
• Correction: Verify assumptions before selecting a statistical test to ensure accurate results.

Quick Practice Problems

1. A company wants to know if their average employee salary exceeds $50,000. A random sample of 36 employees has a mean salary of $52,000 with a standard deviation of $8,000. What is the p-value?

p-value = 0.02 (using a two-tailed t-test with α = 0.05 and df = 35)

1. A marketing firm wants to know if their new campaign increased sales. A sample of 25 stores has a mean sales increase of $1,500 with a standard deviation of $800. What is the critical value for a one-tailed t-test?

Critical Value = t0.05, 24 = 1.711

1. A quality control team wants to know if their manufacturing process meets a certain standard. A sample of 49 products has a mean defect rate of 2% with a standard deviation of 1.5%. What is the p-value for a χ²-test?

p-value = 0.03 (using a χ²-test with α = 0.05 and df = 48)

Last-Minute Cram Sheet

1. Z-test: Use when σ is known and sample size is large (n ≥ 30).
2. t-test: Use when σ is unknown or sample size is small (n < 30).
3. χ²-test: Use for categorical data (e.g., proportions, counts).
4. F-test: Use for comparing variances between groups.
5. p-value: The probability of observing the data (or more extreme) if H₀ is true.
6. α = 0.05: Default significance level.
7. df = n - 1: Degrees of freedom for t-distribution.
8. Critical Value (CV) = Zα/2: For Z-distribution.
9. Critical Value (CV) = tα/2, df: For t-distribution.
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.