Intro to Business Statistics: Random Variables and Probability Distributions Standard Normal Distribution Converting to Z Finding Probabilities Inverse Normal

What This Is

The Standard Normal Distribution is a fundamental concept in statistics used to analyze and interpret data in various business contexts. A retail chain wants to know if average daily sales exceed $10,000, and they collect a random sample of 36 days with a sample mean of $9,800 and a population standard deviation of $500. By converting the sample mean to a Z-score, they can determine the probability of observing such sales or more extreme.

Key Formulas & Symbols

• Z = (x̄ – μ) / (σ/√n) where x̄ = sample mean, μ = population mean, σ = population standard deviation, n = sample size.
• Z-score: A measure of how many standard deviations an observation is from the population mean.
• Standard Normal Distribution: A normal distribution with a mean of 0 and a standard deviation of 1.
• Inverse Normal Distribution: The process of finding the value of x that corresponds to a given Z-score.
• Critical Value: The Z-score that separates the rejection region from the non-rejection region.
• p-value: The probability of observing the data (or more extreme) if the null hypothesis is true.
• Null Hypothesis (H₀): A statement of no effect or no difference.
• Alternative Hypothesis (H₁): A statement of an effect or a difference.

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).
3. Compute test statistic: Calculate the Z-score or t-statistic using the given data.
4. Find p-value or critical value: Determine the p-value or critical value using a standard normal distribution 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 H₀, fail to reject H₀).

Common Mistakes

• Mistake: Using Z when σ is unknown.
• Correction: Use the t-test instead, which is more robust when σ is unknown.

• Why: The t-test takes into account the sample size and the uncertainty of the population standard deviation.

• 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.

• Why: The p-value does not provide information about the probability of H₀ being true.

• Mistake: Failing to check assumptions (e.g., normality, independence).

• Correction: Verify that the data meet the assumptions of the test.
• Why: Failure to meet assumptions can lead to incorrect conclusions.

Quick Practice Problems

1. A company wants to know if the average salary of its employees exceeds $50,000. They collect a random sample of 25 employees with a sample mean of $52,000 and a population standard deviation of $10,000. What is the Z-score?

Answer: Z = (52,000 - 50,000) / (10,000 / √25) = 2.0

Explanation: The Z-score is calculated using the formula Z = (x̄ - μ) / (σ / √n).

1. A marketing firm wants to know if the average response rate to a new advertisement is greater than 10%. They collect a random sample of 100 responses with a sample mean of 12% and a population standard deviation of 5%. What is the p-value?

Answer: p-value = 0.02

Explanation: The p-value is calculated using a standard normal distribution table or calculator, given the Z-score and the significance level (α = 0.05).

1. A quality control team wants to know if the average defect rate of a new manufacturing process is less than 5%. They collect a random sample of 50 products with a sample mean of 4% and a population standard deviation of 2%. What is the confidence interval?

Answer: (3.5%, 4.5%)

Explanation: The confidence interval is calculated using the formula CI = x̄ ± (Z * (σ / √n)), where Z is the critical value for the desired confidence level.

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.
• Z = (x̄ - μ) / (σ / √n) for large samples (n ≥ 30).
• t = (x̄ - μ) / (s / √n) for small samples (n < 30).
• Critical values for Z: Zα/2 = ±1.96 for α = 0.05.
• Critical values for t: tα/2, n-1 = ±2.131 for α = 0.05 and n = 20.
• Inverse Normal Distribution: Z = (x - μ) / σ.
• Standard Normal Distribution: μ = 0, σ = 1.
• Null Hypothesis (H₀): A statement of no effect or no difference.
• Alternative Hypothesis (H₁): A statement of an effect or a difference.
• Significance level (α): The maximum probability of Type I error.