Intro to Business Statistics: Random Variables and Probability Distributions Random Variables Discrete vs Continuous

What This Is

Random variables are a fundamental concept in statistics that help us understand and analyze data in business. A retail chain wants to know if average daily sales exceed $10,000 to determine if they should increase inventory. By understanding discrete and continuous random variables, they can make informed decisions about inventory management, pricing, and marketing strategies.

Key Formulas & Symbols

• Discrete Random Variable (X): A random variable that can only take on distinct, countable values (e.g., number of sales, number of defects).
• Continuous Random Variable (X): A random variable that can take on any value within a given range (e.g., sales amount, temperature).
• Probability Mass Function (PMF): A function that describes the probability of each possible value of a discrete random variable (e.g., P(X = 5) = 0.2).
• Probability Density Function (PDF): A function that describes the probability of each possible value of a continuous random variable (e.g., f(x) = 1/2 for x = 1 to 2).
• Expected Value (E(X)): The long-run average value of a random variable (e.g., E(X) = 10 for a discrete random variable with values 5, 10, and 15).
• Variance (Var(X)): A measure of the spread of a random variable (e.g., Var(X) = 4 for a discrete random variable with values 5, 10, and 15).
• Standard Deviation (σ): The square root of the variance (e.g., σ = √4 = 2 for a discrete random variable with values 5, 10, and 15).
• Normal Distribution: A continuous probability distribution that is symmetric about the mean (e.g., the distribution of sales amounts).
• Z-Score: A standardized value that represents the number of standard deviations from the mean (e.g., Z = (X - μ) / σ).

Step-by-Step Procedure

1. State hypotheses: Clearly define the null and alternative hypotheses (e.g., H₀: μ ≤ 10,000 vs. H₁: μ > 10,000).
2. Choose test: Select an appropriate statistical test based on the type of data and research question (e.g., t-test for comparing means).
3. Compute test statistic: Calculate the test statistic using the sample data and known parameters (e.g., t = (x̄ - μ) / (s / √n)).
4. Find p-value or critical value: Determine the p-value or critical value using a statistical table or software (e.g., p-value = 0.01).
5. Compare to α: Compare the p-value or critical value to the significance level (α = 0.05) to make a decision.
6. Conclude: Based on the comparison, reject or fail to reject the null hypothesis and interpret the results.

Common Mistakes

• Mistake: Using Z when σ is unknown.
• Correction: Use t 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 a test and make adjustments as needed.

Quick Practice Problems

1. A company wants to know if the average daily sales exceed $10,000. The sample mean is $12,000, and the sample standard deviation is $2,000. The sample size is 100. What is the 95% confidence interval for the population mean?

Answer: ($10,000, $14,000) with a margin of error of $2,000.

1. A quality control engineer wants to know if the average number of defects per unit exceeds 5. The sample mean is 6, and the sample standard deviation is 2. The sample size is 50. What is the p-value for the hypothesis test?

Answer: p-value = 0.01.

1. A marketing manager wants to know if the average response time to an advertisement exceeds 30 seconds. The sample mean is 35 seconds, and the sample standard deviation is 10 seconds. The sample size is 200. What is the Z-score for the sample mean?

Answer: Z = (35 - 30) / (10 / √200) = 1.5.

Last-Minute Cram Sheet

1. ⚠️ 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.
2. Use t-test when σ is unknown and sample size is small.
3. Use Z-test when σ is known and sample size is large.
4. Normal distribution is symmetric about the mean.
5. Standard deviation (σ) is the square root of the variance.
6. Expected value (E(X)) is the long-run average value of a random variable.
7. Variance (Var(X)) is a measure of the spread of a random variable.
8. Discrete random variable can only take on distinct, countable values.
9. Continuous random variable can take on any value within a given range.
10. Probability mass function (PMF) describes the probability of each possible value of a discrete random variable.