Question

When Angel commutes to work, the amount of time it takes him to arrive is normally distributed with a mean of 37 minutes and a standard deviation of 3 minutes. Out of the 216 days that Angel commutes to work per year, how many times would his commute be between 31 and 45 minutes, to the nearest whole number?

Answer 1

To find the number of times Angel's commute would be between 31 and 45 minutes, we can use the z-score formula and a standard normal distribution table. The probability of the commute being between 31 and 45 minutes is approximately 0.9544. Therefore, Angel's commute would be between 31 and 45 minutes approximately 206 times out of 216.

Step-by-step explanation:

To find the number of times Angel's commute would be between 31 and 45 minutes, we need to find the area under the normal distribution curve between these two values. We can do this by finding the z-scores for 31 and 45 using the z-score formula: z = (x - mean) / standard deviation. For 31 minutes: z = (31 - 37) / 3 = -2. For 45 minutes: z = (45 - 37) / 3 = 2. Then, we can use the z-scores to find the corresponding probabilities using a standard normal distribution table or a calculator. The probability of the commute being between 31 and 45 minutes is the difference between the probabilities for 31 and 45: P(31 ≤ x ≤ 45) = P(Z ≤ 2) - P(Z ≤ -2).

Using a standard normal distribution table, we find that P(Z ≤ 2) is approximately 0.9772 and P(Z ≤ -2) is approximately 0.0228. Therefore, P(31 ≤ x ≤ 45) is approximately 0.9772 - 0.0228 = 0.9544.

Finally, we can calculate the number of days out of 216 that Angel's commute would be between 31 and 45 minutes by multiplying the probability by 216: 0.9544 * 216 = 206 approximately.