Intro to Business Statistics: Random Variables and Probability Distributions Normal Approximation to Binomial Continuity Correction

What This Is

The Normal Approximation to Binomial (Continuity Correction) is a statistical method used to approximate the binomial distribution with a normal distribution. This is useful when dealing with large sample sizes or when the binomial distribution is difficult to work with. For example, a retail chain wants to know if average daily sales exceed $10,000. They collect data on daily sales over a period of 30 days and want to determine if the average daily sales are significantly higher than $10,000.

Key Formulas & Symbols

• Z = (x̄ – μ) / (σ/√n) where x̄ = sample mean, μ = population mean, σ = population standard deviation, n = sample size.
• p̂ = (x̄ / n) where p̂ = sample proportion, x̄ = sample mean, n = sample size.
• n = N / (1 + (z² / 4)) where n = sample size, N = population size, z = Z-score.
• σ = √(p(1-p) / n) where σ = population standard deviation, p = population proportion, n = sample size.
• Z = (x̄ – np) / √(np(1-p)) where Z = Z-score, x̄ = sample mean, n = sample size, p = population proportion.
• p-value = 2 * P(Z > |Z|) where p-value = probability of observing the data (or more extreme) if H₀ is true, Z = Z-score.
• H₀: p = p₀ where H₀ = null hypothesis, p = population proportion, p₀ = hypothesized population proportion.

Step-by-Step Procedure

1. State hypotheses: State the null and alternative hypotheses in terms of the population proportion (p).
2. Choose test: Choose the Normal Approximation to Binomial (Continuity Correction) test.
3. Compute test statistic: Compute the Z-score using the formula Z = (x̄ – np) / √(np(1-p)).
4. Find p-value or critical value: Find the p-value or critical value using a standard normal distribution table or calculator.
5. Compare to α: Compare the p-value to the significance level (α = 0.05) or compare the test statistic to the critical value.
6. Conclude: Conclude whether to reject the null hypothesis or fail to reject it.

Common Mistakes

• Mistake: Using the Z-score formula when the population standard deviation is unknown.
• Correction: Use the sample standard deviation (s) instead of the population standard deviation (σ) and use the t-distribution instead of the standard normal distribution.
• 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.
• Mistake: Failing to apply the continuity correction when using the Normal Approximation to Binomial.
• Correction: Apply the continuity correction by adding 0.5 to the observed value when using the Z-score formula.

Quick Practice Problems

1. A company wants to know if the average number of defects per unit exceeds 2. They collect data on 100 units and find an average of 2.5 defects per unit. What is the p-value?
2. Answer: 0.012, The p-value is calculated using the Z-score formula and a standard normal distribution table.
3. A marketing firm wants to know if the proportion of customers who respond to a survey is greater than 0.5. They collect data on 500 customers and find that 300 respond. What is the Z-score?
4. Answer: 1.96, The Z-score is calculated using the formula Z = (x̄ – np) / √(np(1-p)).
5. A quality control team wants to know if the average number of defects per unit is less than 1. They collect data on 50 units and find an average of 0.8 defects per unit. What is the p-value?
6. Answer: 0.031, The p-value is calculated using the Z-score formula and a standard normal distribution table.

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) where x̄ = sample mean, μ = population mean, σ = population standard deviation, n = sample size.
• p̂ = (x̄ / n) where p̂ = sample proportion, x̄ = sample mean, n = sample size.
• n = N / (1 + (z² / 4)) where n = sample size, N = population size, z = Z-score.
• σ = √(p(1-p) / n) where σ = population standard deviation, p = population proportion, n = sample size.
• Z = (x̄ – np) / √(np(1-p)) where Z = Z-score, x̄ = sample mean, n = sample size, p = population proportion.
• p-value = 2 * P(Z > |Z|) where p-value = probability of observing the data (or more extreme) if H₀ is true, Z = Z-score.
• H₀: p = p₀ where H₀ = null hypothesis, p = population proportion, p₀ = hypothesized population proportion.
• Use the continuity correction when applying the Normal Approximation to Binomial.
• Use the t-distribution when the population standard deviation is unknown.