// LabEstimationSpecs.txt. 4/10/2020. import Foundation class LabEstimationSpecs { let strEstimationSpecs :String = """ `` Lab Specs KeyWestLows.dat // 0: Population data file 1 2 3 4 10 16 25 50 100 200 400 // 1: List of Sample Sizes .000 .200 .300 .350 .400 .425 .450 .475 .500 // 2: From Values .500 .525 .550 .575 .600 .650 .700 .800 1.000 // 3: To Values `` Prob Specs ` Problem 0 Start Screen 1 - Population Estimates and Random Variables // 0: Title 4 // 1: Sample size T // 2: Pause checkbox selected T F F T // 3: EstMean visible, EstVar visible, ConfInter visible, initial EstMean T T // 4: Set act mean and sample size to estimate variance F // 5: Show To-From .450 .550 // 6: From-To values ` Text Start // 7: Text Two Objectives: Objective 1: Show that the mean of the values sampled in a single sample of the population typically does not equal the actual mean of the population. That is, the estimated population mean, EstMean, typically does not equal the actual population mean, ActMean. Objective 2: Show that the estimated mean is a random variable. Even if we knew the actual population mean, we would not be able to predict the estimated population mean before the population was sampled. By default, the sample size equals 4 and the Pause checkbox is selected. 1. Click Start. What is the estimated mean, EstMean, the mean of the four values randomly selected? Does the estimated mean equal the actual mean? 2. Click Start to simulate a second sample. What is the estimated mean, EstMean, the mean of the four values randomly selected? Does the estimated mean equal the actual mean? 3. Click Start a few more times to simulate more samples. Can we expect the estimated mean to equal the actual mean? 4. Even though we know the actual mean, can we predicted the estimated mean before the sample is conducted? 5. What type of variable is the estimated mean? ` Prob End ` Problem 1 Start Screen 2 - Estimating the Population Mean // 0: Title 1 // 1: Sample size F // 2: Pause checkbox selected T F F T // 3: EstMean visible, EstVar visible, ConfInter visible, initial EstMean T T // 4: Set act mean and sample size to estimate variance F // 5: Show To-From .450 .550 // 6: From-To values ` Text Start // 7: Text Objective: Exploit the relative frequency interpretation of probability to show that the equations we derived for mean and variance of the estimated mean's, EstMean's, probability distribution are correct. Note that the actual population mean, ActMean, equals 64.56 and the actual population variance, ActVar, equals 43.39. Use the equations we dervied to calculate the mean and variance of the estimated mean's, EstMean's, probability distribution when the sample size is 1. 1. 2. 4. 3. 10. Be certain that the Pause checkbox is cleared. Consider each of the sample sizes: 1, 4, and 10. 4. In each case, select the appropriate sample size and click the Start button. After many, many repetitions click the Stop button. How do the mean and variance of EstMean's probability distribution calculated from the equations we derived (questions 1-3) compare to the mean and variance of EstMean's numerical values after many, many repetitions? ` Prob End ` Problem 2 Start Screen 3 - Estimating Population Variance: First Try // 0: Title 4 // 1: Sample size F // 2: Pause checkbox selected F T F F // 3: EstMean visible, EstVar visible, ConfInter visible, initial EstMean T T // 4: Set act mean and sample size to estimate variance F // 5: Show To-From .450 .550 // 6: From-To values ` Text Start // 7: Text Objective: Show that our first estimation procedure for the population variance is unbiased: Calculate the deviation from the actual mean, ActMean. Square the deviation. Sum the squared deviations. Divide the sum of squared deviations by the sample size, T. Note the new fields in the simulation: Use Mean: Act selected to calculate the deviations from actual mean. Divide by: T selected to divide by the sample size. Again the Estimate Variance checkbox is checked; the simulation will be estimated the population variance. By default, the sample size equals 4. Be certain the the Pause checkbox is cleared. 1. Click Start and then after many, many repetitions click Stop. What is the mean (average) of the variance estimates? 2. Is our second estimation procedure for the population variance unbiased? Explain. ` Prob End ` Problem 3 Start Screen 4 - Estimating Population Variance: Second Try // 0: Title 4 // 1: Sample size F // 2: Pause checkbox selected F T F F // 3: EstMean visible, EstVar visible, ConfInter visible, initial EstMean F T // 4: Set act mean and sample size to estimate variance F // 5: Show To-From .450 .550 // 6: From-To values ` Text Start // 7: Text Objective: Show that our second estimation procedure for the population variance is biased: Calculate the deviation from the estimated mean, EstMean. Square the deviation. Sum the squared deviations. Divide the sum of squared deviations by the sample size, T. Note the new fields in the simulation: Use Mean: Est selected to calculate the deviations from estimated mean. Divide by: T selected to divide by the sample size. Again the Estimate Variance checkbox is checked; the simulation will be estimated the population variance. By default, the sample size equals 4. Be certain the the Pause checkbox is cleared. 1. Click Start and then after many, many repetitions click Stop. What is the mean (average) of the variance estimates? 2. Is our second estimation procedure for the population variance unbiased? Explain. ` Prob End ` Problem 4 Start Screen 5 - Estimating Population Variance: Third Try // 0: Title 4 // 1: Sample size F // 2: Pause checkbox selected F T F F // 3: EstMean visible, EstVar visible, ConfInter visible, initial EstMean F F // 4: Set act mean and sample size to estimate variance F // 5: Show To-From .450 .550 // 6: From-To values ` Text Start // 7: Text Objective: Show that our third estimation procedure for the population variance is unbiased: Calculate the deviation from the estimated mean, EstMean. Square the deviation. Sum the squared deviations. Divide the sum of squared deviations by the sample size less 1, T-1. Note the two new fields in the simulation: Use Mean: Est selected to calculate the deviations from estimated mean. Divide by: T-1 selected to divide by the sample size less 1. Again the Estimate Variance checkbox is checked; the simulation will be estimated the population variance. By default, the sample size equals 4. Be certain the the Pause checkbox is cleared. 1. Click Start and then after many, many repetitions click Stop. What is the mean (average) of the variance estimates? 2. Is our third estimation procedure for the population variance unbiased? Explain. ` Prob End ` Problem 5 Start Screen 6 - Confidence Intervals // 0: Title 50 // 1: Sample size T // 2: Pause checkbox selected T F T T // 3: EstMean visible, EstVar visible, ConfInter visible, initial EstMean T T // 4: Set act mean and sample size to estimate variance F // 5: Show To-From .450 .550 // 6: From-To values ` Text Start // 7: Text Objective: Illustrate the meaning of confidence intervals. By default, a 90 percent confidence interval is selected, the sample size equals 50, and the Pause checkbox is selected. Definition: A confidence interval establishes a lower and upper bound for one sampling of the population. In the case of a 90 percent confidence interval, the probability that the actual mean falls between the lower and upper bounds is .90. Two new lines have appeared on the screen: a line reporting on the lower and upper bounds "(LB, UB)" and a line reporting on the repetitions in which the actual mean falls within these bounds. First, let us be certain that the actual means within the bounds are reported correctly. 1. Click Start. What are the lower and upper bounds for the first repetition? For the first repetition, What are the lower and upper bounds? Does the actual mean fall between these bounds? Is the actual mean within the bounds computed correctly for the first repetition? 2. Click Start to simulate a second sample. For the second repetition, What are the lower and upper bounds? Does the actual mean fall between these bounds? Is the actual mean within the bounds computed correctly for the first two repetitions? 3. Click Start to simulate a third sample. For the third repetition, What are the lower and upper bounds? Does the actual mean fall between these bounds? Is the actual mean within the bounds computed correctly for the first three repetitions? 4. Continue doing this until you are convinced that the percent of repetitions in which the actual mean falls within the bounds is being computed correctly. 5. Clear the Pause check box and click Start. After many, many repetitions, click Stop. In what percent of the repetitions does that actual mean fall within the bounds established by the 90 parent confidence interval. ` Prob End """ }