Laboratory Project 5 Confidence Intervals
- Effect of sample size on confidence intervals Introduction: Measuring a statistic in a large population of size N = 1000000 with mean value of 75 and standard deviation with 7.50. Drawing a sample of size n from the population, produces a distribution for the sample mean (X bar) with: E[X bar] = mu_x_bar = mu and E[(X bar – mu_x_bar)^2] = sigma_x_bar^2 Methodology: For case 1 with 95% confidence, using the values of mu +- 1.96 sigma/sqrt(n) as a function of n. For case 2 with 99% confidence, using the values of mu +- 2.58 sigma/sqrt(n) as a function of n.
Result: For case 1 visually confirmed by looking at how many of sample means fall outside of the red curves with approx. of 5%. For case 2 visually confirmed by looking at how many of sample fall outside of the green curves with approx. of 1%.
- Using the sample mean to estimate the population mean
Introduction: To simulate the sample mean to estimate the population mean, generate a barrel of a million ball bearings with weights normally distributed, with a mean with 75 and standard deviation with 7.50. There will be 3 different experiment when n = 5, 40 ,120; using both normal distribution and the t-distribution to complete of the table and compare the success rate between both normal distribution and t-distribution. Methodology: Choose a random sample which is n bearings from 1000000 bearings created in the previous problem and calculate the sample mean and the sample standard deviation.
Check if the confidence interval includes the actual mean 75 µ = of the population of 1,000,000 bearings. If it does, then considered a success. Conclusion: For large sample n > 30 the t-distribution will be very close to normal, so the differences between t-distribution and normal distribution is minimal. Result: Sample size(n) 95% Confidence (Using Normal distribution) 99% Confidence (Using Normal distribution) 95% Confidence (Using Student’s t distribution) 99% Confidence (Using Student’s t distribution) 5 88.25% 95.05% 95.05% 99.19% 40 94.64% 99.12% 95.12% 98.92% 120 95.00% 99.38% 95.17% 99.17%