5.42 The operating times in minutes of three machines are normally dis- tributed with mean times 15, 30 and 20 minutes respectively, and standard deviations 1, 2 and 5 minutes respectively. The machines are independent except for machines 2 and 3 which have operating times that are correlated with r = 0.1. What is the probability that the total operating time of the three machines is at most one hour?

Customers arrive on average every 30 minutes to The Grease Monkey, an auto repair shop with only one mechanic. The inter-arrival times are exponentially distributed. Repair times are variable with a mean of 25 minutes and a standard deviation of 20 minutes. The mechanic works on one vehicle at a time from beginning to end and takes in any waiting vehicles on a first-come first-served basis. The garage itself has room for only one vehicle at a time, so waiting vehicles are kept in the parking lot where there are always plenty of spaces available. Assume customers never balk or renege. a)What is the average number of cars in the garage (not including the parking lot)? b) How long (in minutes) do vehicles wait on average in the adjacent parking lot? c) The competitor shop across the street also has a single mechanic and an average of 3.1 vehicles waiting in its parking lot. The competitor only does oil changes, which take an average of 21 minutes. Customer inter-arrival times to the…

A certain city the Transportation Authority (CTA) has trollies leaving regularly throughout the day beginning at 5:00 a.m. Although the length of a trip and the number of stops are constant, the time taken for each trip varies due to weather, traffic, and time of the day. According to the CTA, the mean time taken for the trip is 50 minutes. Suppose the travel time is Normally distributed with standard deviation o = 4 minutes. A random sample of 25 trips is obtained, and the time for each is recorded. Find the probability that the sample mean time will be less than 48 minutes.

2.7   In China, four-year-olds average three hours a day unsupervised. Most of the unsupervised children live in rural areas, considered safe. Suppose that the standard deviation is 1.5 hours and the amount of time spent alone is normally distributed. We randomly select one Chinese four-year-old living in a rural area. We are interested in the amount of time the child spends alone per day. Find the probability that the child spends less than one hour per day unsupervised. (Leave answer in decimal form rounded to 2 decimal places)

The amount of time that a drive‐through bank teller spends on a customer is a random variablewith a mean of 6.7 minutes and a standard deviation of 3.2 minutes. If a random sample of 64customers is observed, find the probability that their mean time at the teller’s window is: At most 5.7 minutes