STAT240 Homework 6 Solution

Preliminaries
This file should be in STAT240/homework/hw06 on your local computer.
Problem 1
For each of the following questions, say whether the random variable is reasonably approximated by a binomial random variable or not, and explain your answer. If the variable is binomial, identify (n) the number of trials and (p) the probability of success. If it is not binomial, identify which of the “BINS” assumptions is violated.
a. A fair die is rolled until a 1 appears, and (X) denotes the number of rolls.
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b. Twenty of the different Badger basketball players each attempt 1 free throw and (X) is the total number of successful attempts.
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c. A die is rolled 50 times. Let (X) be the face that lands up most often.
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d. In a bag of 10 batteries, I know 2 are old. Let (X) be the number of old batteries I choose when taking a sample of 4 to put into my calculator.
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e. It is reported that 20% of Madison homeowners have installed a home security system. Let (X) be the number of homes without home security systems installed in a random sample of 100 houses in the Madison city limits.
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Problem 2
Create a data frame with the following columns. Each row corresponds to a single ( ext{Binom}(n, p)) distribution. The first two columns are the parameters of the distribution.

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Problem 3
The random variable (X) has the ( ext{Binom}(100, 0.2)) distribution.
Find an integer (a) so that (P(X le a) ge 0.5) and (P(X ge a) ge 0.5). Show the values of (a), (P(X le a)), and (P(X ge a)).
Problem 4
A student decided to guess randomly on their True/False quiz. The number of questions they answer correctly is ( ext{Binom}(10, 0.5)). Write code with dbinom , pbinom , or qbinom to calculate that value or probability.
Problem 5
Match the four binomial distributions given below to the appropriate graph in p5_choices.png . Two of the distributions will not be used. Briefly justify your choices.
Binom(12, 0.5)
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Binom(12, 0.6)
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Binom(10, 0.1)
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Binom(10, 0.3)
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Problem 6
Are the following statements true for Binomial distributions, Normal distributions, or both?
This distribution is always symmetric.
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If you know the two parameters of this distribution, you can calculate its mean, any probability, or any quantile.
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If (mu) is the mean of the distribution, then the probability distribution graphically reaches its maximum at (mu).
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If (mu) is the mean of the distribution, then it is possible for the probability of getting exactly (mu) on a random draw to be 0.
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Problem 7
Use pnorm to find the probabilities highlighted below on a N(0, 1) curve.


Problem 8
Let (X_1) and (X_2) be two draws from (X ~ N(10, 4)). Order the five events below based on which events are least to most likely to occur.
Event A: (X_1) > 15
Event B: (X_1) = 15
Event C: (X_1) < 15
Event D: (X_1) > 15 AND (X_2) > 15 Event E: (X_1) > (X_2)
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