2: SETTING UP CONFIDENCE INTERVALS Condence intervals are used in practice frequently for quality control purposes. As an example, you can consider a factory that produces tens of thousands of bottles of soda every day. Suppose that each bottle of soda is supposed to contain exactly 1 kilogram (ie: 1,000 grams) of soda. Since machines are not perfect, not every single bottle will have exactly 1,000 grams of soda. Some of the bottles will have more than 1,000 grams and others will have less than 1,000 grams of soda. Ideally, the factory which produces these bottles of soda would like to measure the amount of soda in every bottle and discard the bottles that have either too much or too little soda. But it would take too much time, effort, and money to measure the amount of soda in every bottle. As a solution to this problem, almost a century ago the British statistician, Ronald Fisher offered the following methodology: Step-l: Take a random sample of a reasonable size. It can be a random sample of 10, 15, 20, 25 bottles etc. from the production line. Step-2: Measure the amount of soda in every bottle in the sample. Using these observations calculate the average and the standard deviation of the amount of soda in the bottles in the sample. Also calculate the standard error of the sample by dividing the standard deviation with the root square of the number of bottles in the sample. Step-3: Calculate the lower bound of the condence interval: Mean of the sample minus (critical value for the desired condence level times the standard error of the sample) Step-4: Calculate the lower bound of the confidence interval: Mean of the sample plus (critical value for the desired confidence level times the standard error of the sample) Step-5: Interpret the condence level that you just set up. With this methodology, if the desired confidence level is 95%, Ronald Fisher says that we can expect 95% of all the bottles filled in the factory to have an amount of soda that will be Step-5: Interpret the confidence level that you just set up. With this methodology, if the desired confidence level is 95%, Ronald Fisher says that we can expect 95% of all the bottles filled in the factory to have an amount of soda that will be between the lower bound and the upper bound of the confidence interval which you calculated. For a picture of Ronald Fisher, see Figure-3 below: Figure-3: A young Ronald Fisher in 1913. The F distribution in statistics is named after Fisher One thing we should keep in mind is that the confidence interval we set up is sensitive to the random sample of bottles we picked up from the production line. Hence, with each different random sample, we would calculate a different lower bound, and a different upper bound for the confidence interval. This will be important later. Please make sure to keep it in mind.Now we will make use of the information above. 21. Suppose you pick a random sample of 15 bottles of soda from the production line in a soda factory. The bottles in the sample contain the following amounts of soda: Bottle 1: 975 grams Bottle 2: 985 grams Bottle 3'. 990 grams Bottle 4: 981 grams Bottle 5: 996 grams Bottle 6: 1014 grams Bottle 7: 998 grams Bottle 8: 1004 grams Bottle 9: 1024 grams Bottle 10: 995 grams Bottle 11: 1002 grams Bottle 12: 1000 grams Bottle 13: 999 grams Bottle 14: 969 grams Bottle 15: 1018 grams Enter this data into Excel. Using Excel, calculate the average amount of soda in the bottles in this sample. Next calculate the standard deviation of the amount of soda in the bottles in the sample using Excel. Finally, divide this standard deviation with the root square of 15 (ie: sample size) to calculate the standard error of the sample again using Excel. b. Assume a 95% condence interval. The critical value we can use to set up the 95% condence interval is 1.96. Calculate the lower bound and upper bound of the condence interval. c. After you set up the condence interval, interpret the result