--- title: "Sample mean, variance, CV: patients' sleep hours" subject: "Probability and Statistics" chapter: 1 key: "semester_5/probability_and_statistics:hospital-patients-sleep-mean-cv" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/hospital-patients-sleep-mean-cv site: BCT Engineering --- # Sample mean, variance, CV: patients' sleep hours Probability and Statistics · Chapter 1: Descriptive Statistics and Basic Probability - Asked in: 2073 Shrawan, 8 marks; 2072 Kartik, 8 marks; 2070 Asar, 8 marks (asked 3 times) - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/hospital-patients-sleep-mean-cv - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1 ## Question The following table shows the number of hours 45 hospital patients slept following the administration of a certain anesthetic. 7 10 12 4 8 7 3 8 5 12 11 3 8 1 1 13 10 4 4 5 5 8 7 7 3 2 3 8 13 1 7 17 3 4 5 5 3 1 17 10 4 7 7 11 8 a) Find sample mean, sample variance and sample standard deviation. b) Compute a value that measures the amount of variability relative to the value of mean. ## Short answer Sample mean = 6.71 hours, variance = 16.21 hours², standard deviation = 4.03 hours. The coefficient of variation measuring relative variability is (4.03/6.71)×100% = 60%. ## Answer The sample mean is x̄ = Σx / n, the sample variance is s² = Σ(x − x̄)² / (n − 1), the sample standard deviation is s = √s², and the coefficient of variation is CV = (s / x̄) × 100%. #### a) Mean, variance and standard deviation Data: n = 45 values. - Σx = 302 - Σx² = 2740 Mean: ```text x̄ = Σx / n = 302 / 45 = 6.711 hours ``` Sum of squares about the mean (corrected sum of squares): ```text Sxx = Σx² − (Σx)²/n = 2740 − (302)²/45 = 2740 − 2026.756 = 713.244 ``` Variance and standard deviation: ```text s² = Sxx / (n − 1) = 713.244 / 44 = 16.210 hours² s = √16.210 = 4.026 hours ``` #### b) Relative variability The measure that compares spread with the mean is the **coefficient of variation**: ```text CV = (s / x̄) × 100 = (4.026 / 6.711) × 100 = 59.99 % ``` **Answer:** x̄ ≈ 6.71 hours, s² ≈ 16.21 hours², s ≈ 4.03 hours, CV ≈ 60 %. The standard deviation is about 60% of the mean, so the sleeping hours vary a lot between patients. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is the sample variance for the 45 patients found by dividing Sxx by 44? A. Sample variance uses n - 1 = 44 B. It uses n = 45 C. It uses n + 1 = 46 D. It uses 2n Answer: A. For a sample, the divisor is n - 1, so s^2 = 713.244/44. (Key point: Sample variance divisor) 2. Which formula gives Sxx in the answer? A. Sxx = Σx^2 - Σx B. Sxx = Σx^2 - (Σx)^2/n C. Sxx = (Σx)^2 - Σx^2/n D. Sxx = Σx^2/n - x̄ Answer: B. Sxx = Σx^2 - (Σx)^2/n = 2740 - 2026.756 = 713.244. (Key point: Corrected sum of squares) 3. Which measure expresses variability relative to the mean? A. Range B. Variance C. Coefficient of variation = (s/x̄) x 100% D. Standard error Answer: C. CV divides s by the mean, giving a unit-free percentage. (Key point: Coefficient of variation) 4. The answer finds s = 4.026 and x̄ = 6.711. What is the CV and what does it show? A. About 16%; spread is small B. About 0.6 hours; spread is moderate C. About 4%; data are nearly constant D. About 60%; spread is large compared with the mean Answer: D. CV = 4.026/6.711 x 100 = about 60%, so sleep hours vary a lot. 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