Final answer:
Using Chebyshev's theorem and the empirical rule, we can calculate the minimum percentage of individuals who sleep between certain hours per day.
Step-by-step explanation:
(a) To calculate the minimum percentage of individuals who sleep between 2.5 and 9.7 hours using Chebyshev's theorem, we need to determine the range of hours within 2 and 3 standard deviations from the mean. The minimum percentage is given by 1 - 1/k^2, where k is the number of standard deviations. In this case, the range is 6.1 - 2(1.8) = 2.5 to 6.1 + 2(1.8) = 9.7. Hence, k = 2.5/1.8 = 1.39. Therefore, the minimum percentage of individuals who sleep between 2.5 and 9.7 hours is 1 - 1/(1.39)^2 = 1 - 1/1.93 = 1 - 0.5181 = 0.4819 or 48.19%.
(b) Using the same formula, the range of hours within 2 standard deviations from the mean is 6.1 - 2(1.8) = 2.5 to 6.1 + 2(1.8) = 9.7. Therefore, the minimum percentage of individuals who sleep between 1.6 and 10.6 hours is 1 - 1/(2.5/1.8)^2 = 1 - 1/3.0864 = 1 - 0.3245 = 0.6755 or 67.55%.
(c) The empirical rule states that for a bell-shaped distribution, approximately 68% of values lie within one standard deviation of the mean, 95% lie within two standard deviations, and 99.7% lie within three standard deviations. Since the range of hours is within two standard deviations from the mean, we can reasonably estimate that around 95% of individuals sleep between 2.5 and 9.7 hours per day.