Intro to Business Statistics: Sampling and Sampling Distributions - Sampling Distribution of the, Sample Mean Mean of Sample Means Standard Error Shape

What This Is

The sampling distribution of the sample mean is a fundamental concept in statistics that helps businesses make informed decisions. A retail chain wants to know if average daily sales exceed $10,000. By analyzing the sampling distribution of the sample mean, they can determine if their sample mean sales are significantly higher than the population mean, and make decisions about inventory, staffing, and marketing.

Key Formulas & Symbols

• ?x? = ? where ?x? = population mean of sample means,-= population mean.
• ?x? =-/ ?n where ?x? = standard error of the sample mean,-= population standard deviation, n = sample size.
• Z = (x? – ?) / (?/?n) where Z = test statistic, x? = sample mean,-= population mean,-= population standard deviation, n = sample size.
• t = (x? – ?) / (s / ?n) where t = test statistic, x? = sample mean,-= population mean, s = sample standard deviation, n = sample size.
• t-distribution: a probability distribution used for small sample sizes (n < 30).
• Z-distribution: a probability distribution used for large sample sizes (n-30).
• df = n - 1 where df = degrees of freedom, n = sample size.
• p-value: the probability of observing the data (or more extreme) if the null hypothesis is true.

Step-by-Step Procedure

1. State hypotheses: Define the null and alternative hypotheses (e.g., H?:-= 10,000 vs. H?:-> 10,000).
2. Choose test: Select the appropriate test statistic (Z or t) based on the sample size and population standard deviation.
3. Compute test statistic: Calculate the test statistic using the sample mean, population mean, and standard error.
4. Find p-value or critical value: Determine the p-value or critical value from the Z or t-distribution.
5. Compare to ?: Compare the p-value or critical value to the significance level (? = 0.05).
6. Conclude: Make a decision based on the comparison (e.g., reject H? if p-value < ?).

Common Mistakes

• Mistake: Using Z when-is unknown.
• Correction: Use t instead, as it is more robust to small sample sizes and unknown population standard deviation.
• 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: Verify that the assumptions are met before proceeding with the analysis.

Quick Practice Problems

1. A marketing firm wants to know if the average response rate to their email campaign is higher than 5%. They collect a sample of 100 responses with a mean of 6% and a standard deviation of 2%. What is the 95% confidence interval for the population mean response rate? Answer: (5.42, 6.58) The confidence interval is calculated using the sample mean, standard error, and critical value from the Z-distribution.
2. A quality control engineer wants to determine if the average defect rate in a manufacturing process is higher than 2%. They collect a sample of 25 defects with a mean of 3% and a standard deviation of 1.5%. What is the p-value for the null hypothesis that the population mean defect rate is 2%? Answer: 0.012 The p-value is calculated using the test statistic, degrees of freedom, and t-distribution.
3. A sales manager wants to know if the average sales revenue per customer is higher than $100. They collect a sample of 50 customers with a mean of $120 and a standard deviation of $20. What is the Z-score for the sample mean? Answer: 2.24 The Z-score is calculated using the sample mean, population mean, and standard error.

Last-Minute Cram Sheet

1. Z = (x? – ?) / (?/?n): test statistic for large sample sizes.
2. t = (x? – ?) / (s / ?n): test statistic for small sample sizes.
3. t-distribution: used for small sample sizes (n < 30).
4. Z-distribution: used for large sample sizes (n-30).
5. df = n - 1: degrees of freedom for t-distribution.
6. p-value: probability of observing the data (or more extreme) if H? is true.
7. ? = 0.05: default significance level.
8. 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.
9. Use t instead of Z when-is unknown.
10. Verify assumptions (normality, equal variances) before proceeding with analysis.