Intro to Business Statistics: Sampling and Sampling Distributions - Central Limit, Theorem CLT for Means and Proportions

What This Is

The Central Limit Theorem (CLT) is a fundamental concept in statistics that allows us to make inferences about a population based on a sample. It states that the distribution of sample means will be approximately normal, regardless of the population distribution, as long as the sample size is sufficiently large. This is crucial in business decisions, such as determining the average daily sales of a retail chain, the average return on investment of a portfolio, or the average quality of a manufacturing process.

Key Formulas & Symbols

• CLT for Means: x? ~ N(?, ?²/n) where x? = sample mean,-= population mean,-= population standard deviation, n = sample size.
• CLT for Proportions: p? ~ N(p, p(1-p)/n) where p? = sample proportion, p = population proportion, n = sample size.
• Z = (x? – ?) / (?/?n) where x? = sample mean,-= population mean,-= population standard deviation, n = sample size.
• t = (x? – ?) / (s/?n) where x? = sample mean,-= population mean, s = sample standard deviation, n = sample size.
• t = (p? – p) / (?(p(1-p)/n)) where p? = sample proportion, p = population proportion, n = sample size.
• ?² = (n-1)s² / ?² where s² = sample variance, ?² = population variance, n = sample size.
• df = n-1 where n = sample size.
• ? = 0.05 (default significance level).
• Z?/2 = 1.96 (critical value for two-tailed test at-= 0.05).
• t?/2 = 2.045 (critical value for two-tailed test at-= 0.05 with 20 degrees of freedom).

Step-by-Step Procedure

1. State hypotheses: Clearly define the null and alternative hypotheses (H? and H?).
2. Choose test: Select the appropriate test statistic (Z or t) based on the population standard deviation (?) and sample size (n).
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 from the test statistic distribution (Z or t).
5. Compare to ?: Compare the p-value or critical value to the significance level (?).
6. Conclude: Make a decision based on the comparison (reject H? or fail to reject H?).

Common Mistakes

• Mistake: Using Z when-is unknown.
• Correction: Use t instead, as it is more robust to sample variability.
• 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.
• Mistake: Failing to check assumptions (normality, independence).
• Correction: Verify that the data meet the assumptions before applying the CLT.

Quick Practice Problems

1. A retail chain wants to know if average daily sales exceed $10,000. A random sample of 36 days yields a mean sales of $11,500 with a standard deviation of $2,000. Calculate the 95% confidence interval for the population mean. Answer: ($10,419.19, $12,580.81). The calculation involves using the Z-score for a 95% confidence interval.
2. A marketing firm wants to determine if the proportion of customers who prefer a new product is greater than 0.5. A random sample of 100 customers yields 62 who prefer the product. What is the p-value for testing H?: p = 0.5 against H?: p > 0.5? Answer: 0.012. The calculation involves using the Z-score for a one-tailed test.
3. A manufacturing process has a mean quality rating of 80 with a standard deviation of 5. A random sample of 25 products yields a mean quality rating of 85 with a sample standard deviation of 3. What is the p-value for testing H?:-= 80 against H?:-? 80? Answer: 0.011. The calculation involves using the t-score for a two-tailed test.

Last-Minute Cram Sheet

• 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.
• Z = (x? – ?) / (?/?n) for known ?.
• t = (x? – ?) / (s/?n) for unknown ?.
• t = (p? – p) / (?(p(1-p)/n)) for proportions.
• ?² = (n-1)s² / ?² for variance.
• df = n-1 for t-distribution.
• ? = 0.05 (default significance level).
• Z?/2 = 1.96 (critical value for two-tailed test at-= 0.05).
• t?/2 = 2.045 (critical value for two-tailed test at-= 0.05 with 20 degrees of freedom).
• CLT for Means: x? ~ N(?, ?²/n).
• CLT for Proportions: p? ~ N(p, p(1-p)/n).