Intro to Business Statistics: Correlation and Regression - Confidence and Prediction, Intervals Mean Response vs. Individual Response

What This Is

Confidence and prediction intervals are statistical tools used to estimate a population parameter or a future outcome based on a sample of data. A retail chain wants to know if the average daily sales of its new store will exceed $10,000. To answer this question, the chain's management can use a confidence interval to estimate the population mean daily sales.

Key Formulas & Symbols

• Confidence Interval (CI) for Population Mean: CI = x? ± (Z * (?/?n)) where x? = sample mean, Z = critical value from standard normal distribution,-= population standard deviation, n = sample size.
• Margin of Error (ME): ME = Z * (?/?n) where Z = critical value from standard normal distribution,-= population standard deviation, n = sample size.
• Prediction Interval (PI) for Individual Response: PI = x? ± (t * (?/?n)) where x? = sample mean, t = critical value from t-distribution,-= population standard deviation, n = sample size, and df = n - 1.
• Z-score: Z = (x? – ?) / (?/?n) where x? = sample mean,-= population mean,-= population standard deviation, n = sample size.
• t-score: t = (x? – ?) / (?/?n) where x? = sample mean,-= population mean,-= population standard deviation, n = sample size, and df = n - 1.
• Population Standard Deviation (?): a measure of the spread or dispersion of the population.
• Sample Size (n): the number of observations in the sample.
• Critical Value (Z or t): a value from the standard normal distribution or t-distribution that is used to determine the margin of error.
• Degrees of Freedom (df): the number of observations in the sample minus one.

Step-by-Step Procedure

1. State Hypotheses: Define the null and alternative hypotheses. For example, H?:-= 10,000 (the average daily sales is equal to $10,000) and H?:-> 10,000 (the average daily sales is greater than $10,000).
2. Choose Test: Select the appropriate test, either a confidence interval or a prediction interval, depending on the research question.
3. Compute Test Statistic: Calculate the Z-score or t-score using the sample mean, population mean, population standard deviation, and sample size.
4. Find p-value or Critical Value: Determine the p-value or critical value from the standard normal distribution or t-distribution using the calculated test statistic.
5. Compare to ?: Compare the p-value or critical value to the significance level (? = 0.05).
6. Conclude: Based on the comparison, accept or reject the null hypothesis and interpret the results.

Common Mistakes

• Mistake: Using Z when-is unknown.
• Correction: Use t instead of Z when the population standard deviation is unknown.
• Mistake: Misinterpreting p-value as probability H? is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if H? is true, not the probability that H? is true.
• Mistake: Failing to check assumptions (e.g., normality, equal variances).
• Correction: Check assumptions before selecting the test and interpreting results.

Quick Practice Problems

1. A company wants to estimate the average daily production of its new factory. A sample of 36 days has a mean production of 250 units with a population standard deviation of 15 units. Calculate the 95% confidence interval for the population mean. Answer: CI = 250 ± (1.96 * (15/?36)) = 250 ± 6.25, which translates to (243.75, 256.25).
2. A marketing firm wants to predict the sales of a new product. A sample of 25 customers has a mean sales of $100 with a population standard deviation of $20. Calculate the 90% prediction interval for an individual customer's sales. Answer: PI = 100 ± (1.729 * (20/?25)) = 100 ± 7.29, which translates to (92.71, 107.29).
3. A quality control team wants to determine if the average weight of a new product exceeds 10 pounds. A sample of 49 products has a mean weight of 11 pounds with a population standard deviation of 2 pounds. What is the p-value for the test? Answer: p-value = 0.001, which is less than-= 0.05, so we reject the null hypothesis.

Last-Minute Cram Sheet

1. CI = x? ± (Z * (?/?n)) where x? = sample mean, Z = critical value from standard normal distribution,-= population standard deviation, n = sample size.
2. ME = Z * (?/?n) where Z = critical value from standard normal distribution,-= population standard deviation, n = sample size.
3. PI = x? ± (t * (?/?n)) where x? = sample mean, t = critical value from t-distribution,-= population standard deviation, n = sample size, and df = n - 1.
4. Z = (x? – ?) / (?/?n) where x? = sample mean,-= population mean,-= population standard deviation, n = sample size.
5. t = (x? – ?) / (?/?n) where x? = sample mean,-= population mean,-= population standard deviation, n = sample size, and df = n - 1.
6. ? is the population standard deviation.
7. n is the sample size.
8. Z or t is the critical value from the standard normal distribution or t-distribution.
9. df is the degrees of freedom (n - 1).
10. p-value is NOT the probability that H? is true – it’s the probability of observing the data (or more extreme) if H? is true.