Intro to Business Statistics: Random Variables and Probability Distributions Probability Density Function for Continuous Variables

What This Is

The probability density function (PDF) for continuous variables is a fundamental concept in statistics that helps us understand the distribution of data in business. For instance, a retail chain wants to know if average daily sales exceed $10,000. By using the PDF, we can calculate the probability of observing sales above a certain threshold, which informs business decisions such as inventory management and pricing strategies.

Key Formulas & Symbols

• Probability Density Function (PDF): f(x) = 1 / (b - a) where a and b are the lower and upper bounds of the interval.
• Uniform Distribution: f(x) = 1 / (b - a) where a and b are the lower and upper bounds of the interval.
• Exponential Distribution: f(x) = λe^(-λx) where λ is the rate parameter.
• Normal Distribution: f(x) = (1 / (σ√(2π))) * e^(-((x - μ)^2) / (2σ^2)) where μ is the mean and σ is the standard deviation.
• Standard Normal Distribution: f(x) = (1 / √(2π)) * e^(-x^2 / 2).
• Cumulative Distribution Function (CDF): F(x) = ∫[a, x] f(t) dt.
• Inverse Cumulative Distribution Function (ICDF): x = F^(-1)(p) where p is the probability.
• Z-score: Z = (x - μ) / (σ / √n) where x is the value, μ is the mean, σ is the standard deviation, and n is the sample size.
• T-score: T = (x̄ - μ) / (s / √n) where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.

Step-by-Step Procedure

1. State hypotheses: Clearly define the null and alternative hypotheses.
2. Choose test: Select the appropriate statistical test based on the research question and data distribution.
3. Compute test statistic: Calculate the test statistic using the chosen formula (e.g., Z-score or T-score).
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 = 0.05).
6. Conclude: Make a decision based on the comparison (e.g., reject the null hypothesis if p-value < α).

Common Mistakes

• Mistake: Misinterpreting the p-value as the probability that the null hypothesis is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if the null hypothesis is true. It does not provide information about the probability of the null hypothesis being true.
• Mistake: Failing to check assumptions before selecting a statistical test.
• Correction: Always check assumptions such as normality, independence, and equal variances before selecting a statistical test.
• Mistake: Using the wrong distribution (e.g., using a normal distribution when the data is skewed).
• Correction: Choose the distribution that best fits the data and research question.

Quick Practice Problems

1. A marketing firm wants to know if the average response time to an ad is less than 2 minutes. The sample mean is 1.8 minutes, and the sample standard deviation is 0.5 minutes. The sample size is 100. What is the Z-score?

Z = (1.8 - 2) / (0.5 / √100) = -0.2 / 0.05 = -4

The Z-score is -4.

1. A quality control team wants to know if the average defect rate is greater than 5%. The sample mean is 6%, and the sample standard deviation is 2%. The sample size is 50. What is the p-value?

Using a T-distribution table with 49 degrees of freedom (n-1), we find the p-value to be approximately 0.01.

The p-value is 0.01.

1. A sales team wants to know if the average sales revenue is greater than $10,000. The sample mean is $12,000, and the sample standard deviation is $2,000. The sample size is 20. What is the confidence interval?

Using a T-distribution table with 19 degrees of freedom (n-1), we find the confidence interval to be ($9,419, $14,581).

The confidence interval is ($9,419, $14,581).

Last-Minute Cram Sheet

1. Normal Distribution: f(x) = (1 / (σ√(2π))) * e^(-((x - μ)^2) / (2σ^2)).
2. Standard Normal Distribution: f(x) = (1 / √(2π)) * e^(-x^2 / 2).
3. Z-score: Z = (x - μ) / (σ / √n).
4. T-score: T = (x̄ - μ) / (s / √n).
5. p-value: The probability of observing the data (or more extreme) if the null hypothesis is true.
6. α = 0.05: The default significance level.
7. ⚠️ 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.
8. ⚠️ Always check assumptions before selecting a statistical test.
9. ⚠️ Choose the distribution that best fits the data and research question.
10. ⚠️ Use the correct formula for the test statistic (e.g., Z-score or T-score).