Description
Problem 1 (1 point)
Suppose we flip a fair coin 10 times. What is the probability of observing seven or more
heads (Event A)? What is the probability of observing three or less heads (Event B)?
Which event is more likely to happen? Show your derivations and round your numeric
answer to 2 decimal places.
Problem 2 (1 × 3 = 3 points)
Let’s play the Monopoly board game. Each player needs to roll a pair of dice and move
his/her game piece the same number of spaces. Suppose you rolled a pair of dice 100 times
before the end of the game. Let X be the total value shown on the two dice.
• Use the function sample() to simulate the process (i.e. generate 100 realizations of X;
before your implementation, run set.seed(20220404) to set the seed of R’s random
number generator so that the simulation can be reproduced).
• Use the function ggplot() to plot the histogram of those 100 values of x.
• Write down the formulae to calculate sample mean and the sample variance. Show
your numeric answers by using R code and round your numeric answer to 2 decimal
places.
Problem 3 (1.5 × 2 = 3 points)
Currently, there are 52 enrolled students in STAT 3355. It is known that 13.1% of the
population in U.S. are left-handed.
• Use related d, p, q, and r functions to calculate the probability that 10 or fewer lefthanded students in this class?
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• Use the function ggplot() to plot a bar chart, where x axis represents the number of
left-handed students between 0 and 20, while y represent the corresponding probabilities of observation such a number of left-handed students.
Problem 4 (1.5 × 2 = 3 points)
Cereal is sold by weight not volume. This introduces variability in the volume due to
settling. As such, the height to which a cereal box is filled is random. Suppose the heights
for a certain type of cereal box have a normal distribution with mean 12 and standard
deviation 0.5 in units of inches.
• Use related d, p, q, and r functions to calculate the probability that a randomly chosen
cereal box has heights of 10.7 inches or less?
• Use related d, p, q, and r functions to find the lower-quartile, median, and upperquartile of the normal distribution used in this problem. Round your numeric answer
to 2 decimal places.
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