The number of hours people spend traveling for the holidays is normally distributed, with a mean of 4.5 hours and a standard deviation of 0.75 hours. Find each of the following probabilities using a graphing calculator.
a) What is the probability that a randomly selected person spends less than 3 hours traveling for the holidays?
Using a graphing calculator, we can find this probability by calculating the z-score for 3 hours using the formula:
z = (X - μ) / σ
z = (3 - 4.5) / 0.75
z = -2
Then, we look up the z-score in a standard normal distribution table or use the calculator to find the probability. The probability of z = -2 is approximately 0.0228.
Therefore, the probability that a randomly selected person spends less than 3 hours traveling for the holidays is 0.0228 or 2.28%.
b) What is the probability that a randomly selected person spends between 2 and 5 hours traveling for the holidays?
To find this probability, we need to calculate the z-scores for 2 and 5 hours and then find the area between these two z-scores in the standard normal distribution table or by using a graphing calculator.
For 2 hours:
z = (2 - 4.5) / 0.75
z = -3.33
For 5 hours:
z = (5 - 4.5) / 0.75
z = 0.67
Using a graphing calculator, we find the probability of z = -3.33 is approximately 0.0004 and the probability of z = 0.67 is approximately 0.7486.
Therefore, the probability that a randomly selected person spends between 2 and 5 hours traveling for the holidays is 0.7486 - 0.0004 = 0.7482 or 74.82%.