Intro to Business Statistics: Sampling and Sampling Distributions Sampling Error vs Nonsampling Error

What This Is

Sampling error and non-sampling error are two types of errors that can occur when collecting and analyzing data. Sampling error occurs when a sample is not representative of the population, resulting in an estimate that is different from the true population parameter. Non-sampling error, on the other hand, occurs due to factors such as measurement errors, non-response, or data entry errors. A retail chain wants to know if average daily sales exceed $10,000. They collect a sample of 36 days and find the sample mean to be $9,800. However, the sample is not representative of the population, and the estimate is different from the true population parameter.

Key Formulas & Symbols

• Sampling Error: The difference between the sample estimate and the true population parameter.
• Non-Sampling Error: Errors that occur due to factors such as measurement errors, non-response, or data entry errors.
• Bias: A systematic error that occurs when the sample is not representative of the population.
• Variance: A measure of the spread of the data.
• Standard Deviation: The square root of the variance.
• Coefficient of Variation: The ratio of the standard deviation to the mean.
• Confidence Interval: A range of values within which the true population parameter is likely to lie.
• Margin of Error: The maximum amount by which the sample estimate may differ from the true population parameter.
• Z-Score: A measure of how many standard deviations an observation is from the mean.
• p-Value: The probability of observing the data (or more extreme) if the null hypothesis is true.

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 type.
3. Compute Test Statistic: Calculate the test statistic using the sample data and population parameters.
4. Find p-Value or Critical Value: Determine the p-value or critical value using the test statistic and the chosen test.
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 and state the conclusion.

Common Mistakes

• Mistake: Using Z when σ is unknown.
• Correction: Use t-statistic when σ is unknown, and the sample size is small (n < 30).
• 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.
• Mistake: Failing to check assumptions.
• Correction: Check assumptions such as normality, independence, and equal variances before selecting the test.

Quick Practice Problems

1. A company wants to estimate the average salary of its employees. They collect a sample of 25 employees and find the sample mean to be $50,000. The population standard deviation is $10,000. What is the margin of error?

Answer: $2,500. The margin of error is calculated using the formula: Margin of Error = (Z * σ) / √n, where Z is the Z-score, σ is the population standard deviation, and n is the sample size.

1. A marketing firm wants to know if the average daily sales exceed $10,000. They collect a sample of 36 days and find the sample mean to be $9,800. The sample standard deviation is $2,000. What is the p-value?

Answer: 0.02. The p-value is calculated using the t-test, and the degrees of freedom are n-1 = 35.

1. A quality control team wants to estimate the proportion of defective products. They collect a sample of 100 products and find 10 defective products. What is the confidence interval?

Answer: (0.05, 0.15). The confidence interval is calculated using the formula: CI = (p̂ - (Z * √(p̂(1-p̂)/n)), p̂ + (Z * √(p̂(1-p̂)/n))), where p̂ is the sample proportion, Z is the Z-score, and n is the sample size.

Last-Minute Cram Sheet

1. Sampling Error: The difference between the sample estimate and the true population parameter.
2. Non-Sampling Error: Errors that occur due to factors such as measurement errors, non-response, or data entry errors.
3. Bias: A systematic error that occurs when the sample is not representative of the population.
4. Variance: A measure of the spread of the data.
5. Standard Deviation: The square root of the variance.
6. Coefficient of Variation: The ratio of the standard deviation to the mean.
7. Confidence Interval: A range of values within which the true population parameter is likely to lie.
8. Margin of Error: The maximum amount by which the sample estimate may differ from the true population parameter.
9. Z-Score: A measure of how many standard deviations an observation is from the mean.
10. p-Value: The probability of observing the data (or more extreme) if the null hypothesis is true.
11. α = 0.05: The default significance level.
12. t-Test: Used when σ is unknown and the sample size is small (n < 30).
13. p̂ = (X / n): The sample proportion.
14. CI = (p̂ - (Z * √(p̂(1-p̂)/n)), p̂ + (Z * √(p̂(1-p̂)/n))): The confidence interval formula.
15. ⚠️ 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.