Specialist2023 Exam 1 · Q6Tech-free4 marks

Sum of normal travel times and distribution of a sample mean

Josie travels from home to work in the city. She drives a car to a train station, waits, and then rides on a train to the city. The time, $X_c$ minutes, taken to drive to the station is normally distributed with a mean of 20 minutes ( $\mu_c = 20$ ) and standard deviation of 6 minutes (). The waiting time, minutes, for a train is normally distributed with a mean of 8 minutes () and standard deviation of minutes . The time, minutes, taken to ride on a train to the city is also normally distributed with a mean of 12 minutes () and standard deviation of 5 minutes (). The three times are independent of each other.

Find the mean and standard deviation of the total time, in minutes, it takes for Josie to travel from home to the city.

2 marks

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Josie’s waiting time for a train on each work day is independent of her waiting time for a train on any other work day. The probability that, for 12 randomly chosen work days, Josie’s average waiting time is between 7 minutes 45 seconds and 8 minutes 30 seconds is equivalent to $\Pr(a < Z < b)$ , where $Z \sim \text{N}(0, 1)$ and $a$ and $b$ are real numbers.

Find the values of $a$ and $b$ .

2 marks

Hints, answer checking and worked solutions come with a Season Pass.