Problem 1

The length of a coil of copper wire is a random variable with mean and standard deviation . If we choose five coils of wire at random, what is the variance of the total length of the wire in the coils?

Because each wire is modelled by an independent random variables, their covariances must always be , thus we can use the formula for additions of variances with the covariance being equal.

Problem 2

Which of the following normal distributions has the widest spread?

• [ ] A. A normal distribution with mean 3 and standard deviation 2
• [ ] B. A normal distribution with mean 2 and standard deviation 1
• [x] C. A normal distribution with mean 1 and standard deviation 3
• [ ] D. A normal distribution with mean 0 and standard deviation 2
• [ ] E. None of the above

Problem 3

A random variable follows a normal distribution with mean and standard deviation . Which of the following gives the expectation

• [ ] A. 1936
• [x] B. 1940
• [ ] C. 1932
• [ ] D. 2116
• [ ] E. Insufficient information to calculate

Problem 4

The lengths of a certain type of chain are approximately normally distributed with a mean of and a standard deviation of . Find the value of such that .

• [x] A. 2.80 cm
• [ ] B. 2.23 cm
• [ ] C. 5.19 cm
• [ ] D. 0.30 cm
• [ ] E. 2.10 cm

Searching from the standard normal table we get

Problem 5

A random variable is normally distributed, with a mean of 36 and a standard deviation of 2.2. Which of the following is the appropriate interquartile range for this distribution?

• [ ] A.
• [ ] B.
• [ ] C.
• [ ] D.
• [x] E.

Problem 6

You purchase a chainsaw, and can buy one of two types of batteries to power it, namely Duxcell and Infinitycell. Batteries of each type have lifetimes before recharge that can be assumed independent and normally distributed.

The mean and standard deviation of the lifetimes of the Duxcell batteries are 10 and 2 minutes respectively, the mean and standard deviation for the Infinitycell batteries are 11 and 3 minutes respectively.

a.

What is the probability that a Duxcell battery will last longer than an Infinitycell battery? Give your answer to four decimal places.

Since both and are normally distributed and independent so must also be normally distributed.

Now we need to normalize so that we can use the standard normal table.

b.

What is the probability that an Infinitycell battery will last more than twice as long as a Duxcell battery? Give your answer to four decimal places.

Since both and are normally distributed and independent so must also be normally distributed.

Now we need to normalize so that we can use the standard normal table.

c.

You are going to cut down a large tree and do not want to break off from the job to recharge your chainsaw battery. You buy two Duxcell batteries, and plan to use one until it runs out of power, after which you immediately replace it with the second battery. How long (in minutes) can the job last so that with probability 0.75 you can complete the job using the two Duxcell batteries in sequence? Provide your answer to 1 decimal place.

Problem 7

The time it takes Alice to walk to the bus stop from her home is normally distributed with mean 13 minutes and variance 3 minutes. The waiting time for the bus to arrive is normally distributed with mean 6 minutes and standard deviation 2 minutes.

Her bus journey to the UBC bus loop is a normal variable with mean 24 minutes and standard deviation 5 minutes. The time it take Alice to walk from the bus loop to the lecture theatre to attend STAT 251 is normally distributed with mean 18 minutes and variance 4 minutes. The total time taken for Alice to travel from her home to her STAT 251 lecture is normally distributed. Please use R to find probabilities (R's pnorm() function).

a.

What is the mean travel time (in minutes)?

b.

What is the standard deviation of Alice's travel time (in minutes, to 2 decimal places)?

c.

The STAT 251 class starts at 8 am sharp. Alice leaves home at 7 am. What is the probability (to 2 decimal places) that Alice will not be late for her class?

probability_not_late <- pnorm(max_time, mean = 60, sd = 6.00)
prob_rounded <- round(probability_not_late, 2)

cat(prob_rounded) # 0.43

Problem 8

Suppose a car manufacturer believes its windscreen wipers will last on average for three years on their cars if driven by a typical driver in the province. Moreover, the manufacturer believes the lifetime of the wipers under such conditions is normally distributed with a standard deviation of two years. Find the probability that, if on a car driven by a typical driver, a windscreen wiper lasts for a time that is not within 1.4 years of the mean lifetime.