Final answer:
To find the interval centered at the mean that contains 75% of values for a normal distribution, calculate 1.15 standard deviations from the mean. Having the mean at 49.2 hours and a standard deviation of 4.8 hours, the interval is approximately from 43.68 to 54.72 hours. The closest provided option is c) 44.4 to 54.0 hours.
Step-by-step explanation:
The student is asking about the interval centered at the mean that contains 75% of the values in a normal distribution. We know the mean (μ) is 49.2 hours, and the standard deviation (σ) is 4.8 hours. To find the interval containing 75% of the values, we will use the property of the normal distribution that approximately 68% of the data lies within one standard deviation of the mean, and approximately 95% lies within two standard deviations.
Since we want 75% which is between 68% and 95%, the interval we seek will be less than two standard deviations from the mean but more than one standard deviation. It is typically the case that approximately 75% will lie within 1.15 standard deviations of the mean for a normal distribution. Therefore, we do the following calculations:
Lower Bound = Mean - (1.15 * Standard Deviation) = 49.2 - (1.15 * 4.8) = 49.2 - 5.52 = 43.68
Upper Bound = Mean + (1.15 * Standard Deviation) = 49.2 + (1.15 * 4.8) = 49.2 + 5.52 = 54.72
The interval that includes approximately 75% of the distribution is therefore from 43.68 to 54.72 hours. The closest option is c) 44.4 to 54.0 hours.