Answer: a) To find the mean and standard deviation of the difference of their times, we need to use the properties of normal distributions. When two independent normal random variables are subtracted, the mean and standard deviation of the difference can be found using the following formulas:
Mean of the difference = Mean of x - Mean of y
Standard deviation of the difference = √(Standard deviation of x^2 + Standard deviation of y^2)
Plugging in the given values:
Mean of the difference = Mean of Molesky - Mean of Liberty = 100 - 110 = -10
Standard deviation of the difference = √(8^2 + 10^2) = √(64 + 100) = √(164) = 12.8
So the mean of the difference of their times is -10 minutes and the standard deviation is 12.8 minutes.
b) To find the probability that Mr. Molesky will finish his game before Mr. Liberty on any given day, we need to find the probability that the difference of their times is less than 0. Since the difference of their times is normally distributed with a mean of -10 and a standard deviation of 12.8, we can use the standard normal distribution table to find this probability.
To calculate the probability, we will standardize the distribution of the difference by subtracting the mean and dividing by the standard deviation.
z = (x - mean) / standard deviation
z = (-10) / 12.8 = -0.78125
Using a standard normal table we can find that P(z<-0.78125) = 0.22 or 22%
So the probability that Mr. Molesky will finish his game before Mr. Liberty on any given day is 22%.
Explanation: