Intro to Business Statistics: Probability Approaches to Probability Classical Empirical Subjective

Approaches to Probability (Classical, Empirical, Subjective)

In business, making informed decisions relies heavily on probability. A retail chain wants to know if average daily sales exceed $10,000 to determine if they should increase inventory. They can use one of three approaches: Classical, Empirical, or Subjective probability.

Key Formulas & Symbols

• Classical Probability: P(A) = (Number of favorable outcomes) / (Total number of outcomes) where A is the event.
• Empirical Probability: P(A) = (Number of times event A occurs) / (Total number of trials).
• Subjective Probability: P(A) = (Degree of belief in event A).
• 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 [(observed frequency – expected frequency)² / expected frequency] where Σ denotes the sum of the squared differences between observed and expected frequencies.
• df = n – 1 where n is the sample size.
• α = 0.05 (default significance level).
• p-value = P(test statistic ≥ observed test statistic) or p-value = P(test statistic ≤ observed test statistic) depending on the direction of the alternative hypothesis.

Step-by-Step Procedure

1. State hypotheses: Clearly define the null (H₀) and alternative (H₁) hypotheses.
2. Choose test: Select the appropriate statistical test based on the type of data and the 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 statistical table or calculator.
5. Compare to α: Compare the p-value or critical value to the significance level (α).
6. Conclude: Based on the comparison, reject or fail to reject the null hypothesis.

Common Mistakes

• Mistake: Misinterpreting the p-value as the probability that H₀ is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if H₀ is true. It does not directly tell us the probability of H₀ being true.
• Mistake: Using the Z-test when the population standard deviation (σ) is unknown.
• Correction: Use the t-test instead, which is robust to small sample sizes and unknown population standard deviation.
• Mistake: Failing to check assumptions before applying a statistical test.
• Correction: Verify that the data meet the necessary assumptions, such as normality and independence, before proceeding with the test.

Quick Practice Problems

1. A marketing firm wants to know if the average response rate to their email campaign exceeds 5%. They randomly sample 36 responses and find a mean response rate of 6.2%. What is the 95% confidence interval for the population mean response rate?

Answer: (5.55, 6.85) Explanation: Use the t-distribution with df = 35 and a two-tailed test.

1. A quality control team wants to determine if the average defect rate in a manufacturing process is greater than 2%. They collect a random sample of 25 products and find a mean defect rate of 2.5%. What is the p-value for the test?

Answer: 0.02 Explanation: Use the Z-test with a one-tailed test and a significance level of 0.05.

1. A sales manager wants to know if the average sales revenue for a new product exceeds $100. They randomly sample 49 sales records and find a mean sales revenue of $120. What is the 99% confidence interval for the population mean sales revenue?

Answer: (114.55, 125.45) Explanation: Use the t-distribution with df = 48 and a two-tailed test.

Last-Minute Cram Sheet

• ⚠️ 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.
• Use the Z-test when σ is known and the sample size is large (n ≥ 30).
• Use the t-test when σ is unknown or the sample size is small (n < 30).
• Verify assumptions before applying a statistical test.
• df = n – 1 for t-tests and χ²-tests.
• α = 0.05 (default significance level).
• p-value = P(test statistic ≥ observed test statistic) or P(test statistic ≤ observed test statistic) depending on the direction of the alternative hypothesis.
• χ²-test is used for categorical data.
• t-test is used for continuous data.
• Z-test is used for large sample sizes and known population standard deviation.