Final answer:
Chebyshev's Inequality can be used to determine what percentage of data falls within a certain range. For part a), no more than 75% of the completion times are between 1.4 hours and 3.4 hours. For part b), at least 75% of the completion times are between 1.4 hours and 3.4 hours. For part c), it is not possible for at most 88.9% of the completion times to be between 1.4 hours and 3.4 hours. And for part d), we can conclude that at least 75% of the completion times are between 1.4 hours and 3.4 hours.
Step-by-step explanation:
Chebyshev's Inequality states that for any data set, regardless of its shape, at least k% of the data values will fall within k% standard deviations of the mean, where k is any positive number greater than 1. In this case, since we want to determine what percentage of the completion times falls within a certain range, we can use Chebyshev's Inequality.
a) To find the percentage of completion times that are between 1.4 hours and 3.4 hours, we need to calculate the number of standard deviations away from the mean these values are. The mean completion time is 2.4 hours, and the standard deviation is 0.5 hours. So, 1.4 hours is (1.4-2.4)/0.5 = -2 standard deviations away from the mean, and 3.4 hours is (3.4-2.4)/0.5 = 2 standard deviations away from the mean. According to Chebyshev's Inequality, at least 75% of the data falls within 2 standard deviations of the mean. Since -2 to 2 standard deviations covers a range of 4 standard deviations, we can conclude that no more than 75% of the completion times are between 1.4 hours and 3.4 hours.
b) Similarly, we can conclude that at least 75% of the completion times are between 1.4 hours and 3.4 hours since this range covers 2 standard deviations away from the mean.
c) To find the percentage of completion times that are between 1.4 hours and 3.4 hours, we need to calculate the number of standard deviations away from the mean these values are. The mean completion time is 2.4 hours, and the standard deviation is 0.5 hours. So, 1.4 hours is (1.4-2.4)/0.5 = -2 standard deviations away from the mean, and 3.4 hours is (3.4-2.4)/0.5 = 2 standard deviations away from the mean. Again, according to Chebyshev's Inequality, at least 75% of the data falls within 2 standard deviations of the mean. Therefore, it is not possible for at most 88.9% of the completion times to be between 1.4 hours and 3.4 hours.
d) Finally, using the same reasoning as in parts a) and b), we can conclude that at least 75% of the completion times are between 1.4 hours and 3.4 hours.