Intro to Business Statistics: Introduction to Statistics What is Statistics Definition Descriptive vs Inferential Statistics

What This Is

Statistics is the science of collecting, analyzing, and interpreting data to make informed business decisions. A retail chain wants to know if average daily sales exceed $10,000 to determine if they should expand their store hours. By using statistics, they can analyze sales data from multiple stores to make a confident decision.

Key Formulas & Symbols

• Descriptive Statistics:
• Mean (x̄): x̄ = (Σx) / n, where x = individual data point, n = sample size.
• Median (M): M = middle value when data is sorted in ascending order.
• Mode (Mo): Mo = most frequently occurring value.
• Range (R): R = maximum value - minimum value.
• Variance (σ²): σ² = Σ(x - x̄)² / (n - 1).

• Standard Deviation (σ): σ = √(σ²).

• Inferential Statistics:

• 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-value: probability of observing the data (or more extreme) if H₀ is true.
• Confidence Interval (CI): CI = x̄ ± (Z * (σ/√n)) or CI = x̄ ± (t * (s/√n)).

Step-by-Step Procedure

1. State hypotheses: H₀ (null hypothesis) and H₁ (alternative hypothesis).
2. Choose test: select the appropriate statistical test (e.g., t-test, ANOVA).
3. Compute test statistic: calculate the test statistic (e.g., Z, t).
4. Find p-value or critical value: determine the p-value or critical value from the test statistic.
5. Compare to α: compare the p-value or critical value to the significance level (α = 0.05).
6. Conclude: reject H₀ if p-value < α or critical value > test statistic.

Common Mistakes

• Mistake: Using Z when σ is unknown.
• Correction: Use t-test instead, as it is more robust to non-normality and can handle unknown σ.
• Why: Z-test assumes σ is known, which is rarely the case in real-world applications.
• Mistake: Misinterpreting p-value as probability H₀ is true.
• Correction: p-value is the probability of observing the data (or more extreme) if H₀ is true.
• Why: p-value does not provide information about the probability of H₀ being true; it only indicates the strength of evidence against H₀.
• Mistake: Failing to check assumptions (e.g., normality, equal variances).
• Correction: Check assumptions before selecting the test and interpreting results.
• Why: Incorrect assumptions can lead to incorrect conclusions and biased results.

Quick Practice Problems

1. A company wants to know if the average salary of their employees is greater than $50,000. They collect a random sample of 25 employees and find a sample mean of $52,000 with a sample standard deviation of $5,000. What is the 95% confidence interval for the population mean?

Answer: CI = $52,000 ± (1.96 * ($5,000/√25)) = $52,000 ± $1,960, or ($50,040, $53,960).
Explanation: The confidence interval is calculated using the sample mean, sample standard deviation, and the critical value from the standard normal distribution (Z = 1.96).

1. A marketing firm wants to know if the average response rate to their email campaign is greater than 10%. They collect a random sample of 100 responses and find a sample proportion of 0.12. What is the p-value for the test?

Answer: p-value = 0.001.
Explanation: The p-value is calculated using the sample proportion, sample size, and the standard normal distribution.

1. A quality control team wants to know if the average defect rate in their manufacturing process is greater than 5%. They collect a random sample of 50 products and find a sample mean of 0.06 with a sample standard deviation of 0.02. What is the test statistic for the t-test?

Answer: t = (0.06 - 0.05) / (0.02/√50) = 1.58.
Explanation: The test statistic is calculated using the sample mean, population mean, sample standard deviation, and sample size.

Last-Minute Cram Sheet

1. Descriptive statistics: summarize data using mean, median, mode, range, variance, and standard deviation.
2. Inferential statistics: make conclusions about a population based on a sample.
3. Z-test: used when σ is known and the sample size is large.
4. t-test: used when σ is unknown or the sample size is small.
5. p-value: probability of observing the data (or more extreme) if H₀ is true.
6. Confidence interval: range of values within which the population parameter is likely to lie.
7. α = 0.05: default significance level for most tests.
8. ⚠️ 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.
9. ⚠️ Always check assumptions (e.g., normality, equal variances) before selecting the test and interpreting results.
10. ⚠️ Use t-test instead of Z-test when σ is unknown or the sample size is small.