Intro to Business Statistics: Statistical Software and Applications Case Studies in Marketing Finance Operations HR

What This Is

Case studies in marketing, finance, operations, and HR involve analyzing data to make informed business decisions. For instance, a retail chain wants to know if average daily sales exceed $10,000 to determine if they should expand their store hours. By applying statistical methods, they can analyze sales data and make a decision based on the results.

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.
• p = 1 - (1 - α)^n where p = probability of success, α = significance level, n = sample size.
• Confidence Interval = x̄ ± (Z * (σ/√n)) where x̄ = sample mean, Z = critical value, σ = population standard deviation, n = sample size.
• Hypothesis Testing: H₀: μ = μ₀ vs. H₁: μ ≠ μ₀ where H₀ = null hypothesis, H₁ = alternative hypothesis, μ = population mean, μ₀ = known population mean.
• Type I Error: α = 0.05 where α = significance level.
• Type II Error: β = 1 - P(Detecting H₁ | H₁ is true) where β = probability of Type II error, P(Detecting H₁ | H₁ is true) = probability of detecting the alternative hypothesis when it is true.

Step-by-Step Procedure

1. State Hypotheses: Clearly define the null and alternative hypotheses.
2. Choose Test: Select the appropriate statistical test (e.g., Z-test, t-test, confidence interval).
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 for the test statistic.
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 results (e.g., reject H₀, fail to reject H₀).

Common Mistakes

• Mistake: Using Z when σ is unknown.
• Correction: Use t-test instead, as it is more robust to non-normality and allows for estimation of σ.
• 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: Check assumptions before selecting the test and interpreting results.

Quick Practice Problems

1. A marketing firm wants to know if the average response rate to their email campaign exceeds 5%. They randomly select 100 responses and find a sample mean of 6%. Assuming a population standard deviation of 2, what is the 95% confidence interval for the population mean? Answer: 5.53, 6.47. The calculation involves using the Z-score for a 95% confidence interval and the given sample mean and standard deviation.
2. A company wants to determine if the average salary of their employees is greater than $50,000. They randomly select 25 employees and find a sample mean of $55,000 with a sample standard deviation of $5,000. What is the p-value for the t-test? Answer: 0.012. The calculation involves using the t-test formula and the given sample mean, standard deviation, and sample size.
3. A retail chain wants to know if the average daily sales exceed $10,000. They randomly select 30 days and find a sample mean of $12,000 with a sample standard deviation of $2,000. What is the 90% confidence interval for the population mean? Answer: 10.53, 13.47. The calculation involves using the Z-score for a 90% confidence interval and the given sample mean and standard deviation.

Last-Minute Cram Sheet

• Z-test: Use when σ is known and sample size is large (n ≥ 30).
• t-test: Use when σ is unknown or sample size is small (n < 30).
• Confidence Interval: Estimate population mean with a specified margin of error.
• p-value: Probability of observing the data (or more extreme) if H₀ is true.
• α = 0.05: Default significance level for most tests.
• Type I Error: Reject H₀ when it is true (α = 0.05).
• Type II Error: Fail to reject H₀ when it is false (β = 1 - P(Detecting H₁ | H₁ is true)).
• Normality Assumption: Data should be approximately normally distributed.
• Equal Variances Assumption: Variances of the populations should be equal.
• ⚠️ 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.