Understanding Central Tendency, Variability, and Z-Scores

Measures of Central Tendency and Variability Emmett Mean: (94 + 74 + 63 + 59 + 74 + 92) / 6 = 76 Mode: 74 Median = (74 + 74) / 2 = 74 Kiernan Mean: (84 + 83 + 73 + 80 + 88 + 80) / 6 = 81.33 Mode: None Median = (80 + 83) / 2 = 81.5 Brennan Mean: (92 + 68 + 69 + 82 + 92 + 78) / 6 = 80.17 Mode: 92 Median = (78 + 82) / 2 = 80 Suggestions for each student Emmett would benefit from using mean. Kiernan should use the median as it is the highest of the 3 methods. Brennan should use mode as the final score as it’s the highest. Measures of Variance and Standard Deviation Class 1: Mean 78.67, Variance 197.75, Std. Dev 14.06 Class 2: Mean 80.56, Variance 62.53, Std. Dev 7.91 Class 3: Mean 85.11, Variance 48.36, Std. Dev 6.95 Class 4: Mean 87.11, Variance 90.11, Std. Dev 9.49 Class 5: Mean 84.44, Variance 138.02, Std. Dev 11.75

No.3 Question Mean: 15.04, Variance: 82.3, Std. Dev: 9.07 Difference between Sample and Population Standard Deviation Sample standard deviation: An estimate of population standard deviation based on a small collection of data (24 airport security guards for this set of data). It approximates the distribution of values in the sample. Population standard deviation: This is an accurate indication of how dispersed the values are throughout the whole population of airport security personnel. It is determined using the same method as the sample standard deviation, but rather than dividing by n-1, divide by n (the whole population size). As a result, the sample standard deviation obtained shows the unpredictability of the security concerns recognized by these specific 24 guards, but the population standard deviation shows the unpredictability of risk identification for all airport security guards. Z table Questions 17 years old candidate Calculate the z-score: z-score = (x - μ) / σ x = 72.1 (graduation rate of the specific flight academy) μ = 80 (mean graduation rate for 17-year-olds) σ = 2.1 (standard deviation for 17-year-olds) z-score = (72.1 - 80) / 2.1 = -3.76 The area is approximately 0.0009, meaning only 0.09% of flight academies have a graduation rate lower than 72.1 for 17-year-olds.

Calculate the percentage with a higher rate: Subtract the area to the left from 100% to find the percentage with a higher rate.100% - 0.09% = 99.91%. Therefore, approximately 99.99% of flight academies have a higher graduation rate for 17-year- old candidates than 72.1. 34 years old candidate Calculate the z-score: z = (83.5 - 83.1) / 1.3 = 0.31 This area is approximately 0.6217. Calculate the percentage with a lower rate: 100% - 62.17% = 37.83% Therefore, approximately 37.83% of flight academies have a lower graduation rate for 34-year- old candidates than 83.5%. 29 years old candidate Calculate the z-score: z = (93.1 - 91.1) / 1.0 = 2 This area is approximately 0.9772. Calculate the percentage below the academy's rate: 100% - 97.72% = 2.28% Since 2.28% of academies have a higher rate than 93.1, the academy is not in the top 10%, so their claim is false. 18-year-old candidate

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The average currently for 18-year-olds is 78.5 %. The 10-year average is higher. Investing in new methods may improve the graduation rate, but I would suggest looking at what caused the fall off for the current year about the 10-year average if the curriculum stayed the same. This decision depends on the academy's definition of better and the cost of the new methods. Cost, time, resources, and other variables should be considered as well. 27 years old candidate Find the z-scores for both periods: 2000-2009: z = (85.1 - 87.5) / 1.9 = -1.26 2010-2018: z = (88.3 - 87.5) / 1.9 = 0.42 2000-2009: Approximately 0.10 2010-2018: Approximately 0.66 Convert these areas to percentiles: 2000-2009: 10th percentile 2010-2018: 66th percentile The academy's graduation rate was below 10% of all academies for 27-year-olds between 200- 2009. From 2010-2018, the academy's graduation rate was above 66% of all academies for 27- year-olds. Based on this change in data, it can be assumed that the change of instructors has worked and the instructions provided are clearer and understood by the students at a better rate. The change in graduation rates changed from the bottom 10% to the top 34% among 27-year- olds.

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