Homework 3
Definitions
Define the following terms/concepts
a.) Sampling Error
b.) Law of Large Numbers
c.) Sample Space
d.) Event
e.) Union
f.) Intersection
g.) Random Trial
h.) Random Variable
i.) Probability distribution
j.) Population distribution
k.) Continuous random variable
l.) Discrete random variable
m.) Probability Mass Function
n.) Probability Density Function
Simple probability
\(\bf (1)\) A fair coin is thrown 3 times. What is the probability that at least one heads is observed?
\(\bf (2)\) Suppose that in a class of 30 students, 8 students are in band, 15 students play a sport, and 5 students are both in band and play a sport. Let \(A\) be the event that a student is in band and let \(B\) be the event that a student plays a sport. Create a Venn diagram that models this situation. What is the probability that a student is in the band or plays a sport?
\(\bf (3)\) A standard deck of 52 cards contains four suits (Hearts, Diamonds, Clubs, and Spades). Each suit contains 13 cards (9 numbered cards and 3 face cards, and an Ace).
- Suppose a single card is drawn at random, what is the probability that it is a numbered?
- What is the probability that the selected card is the Ace of Spades (assume independence where necessary)?
\(\bf (4)\) An urn contains 5 green balls, 7 red balls, and 4 yellow balls. Two balls are selected at random with replacement. What is the probability that first ball is yellow and the second ball is red?
\(\bf (5)\) The probability that it rains \(R\) on a given day in Moscow, Idaho is \(P(R) = 0.1\). The probability that it is Windy \(W\) is \(P(W) = 0.2\), and the probability that it is cloudy \(C\) is \(P(C) = 0.7\). Assume that these three types of weather events are independent. What is the probability that a given day is both Rainy, Cloudy, and Windy?
Sampling Distributions, Means, and Variances
\(\bf (6)\) A random variable \(X\) has the following probability distribution
| X | P(X) |
|---|---|
| 1 | 0.25 |
| 2 | 0.25 |
| 3 | 0.25 |
| 4 | 0.25 |
- Derive the sampling distribution for the mean of \(n=2\) observations of \(X\)
- Compute the population mean \(\mu\) and population standard deviation \(\sigma\) of \(\bar{X}\)
\(\bf (7)\) A random variable \(Y\) has two possible outcomes. The outcome \(Y = 1\) occurs with probability \(p = 0.23\). Derive the sampling distribution for the proportion of outcomes with \(Y = 1\) in a sample of \(n = 3\) observations of \(Y\).
\(\bf (8)\) Consider a survey of tree volume. Suppose that in the population of trees \(10\%\) have a volume of 10 cubic feet, \(20\%\) have a volume of 20 cubic feet, \(30\%\) have a volume of 30 cubic feet, and \(40\%\) have a volume of 40 cubic feet. Let \(X\) denote the volume of a randomly selected tree, assuming that every tree has an equal chance of being selected. The probability distribution of \(X\) is given in the table below.
| X | P(X) |
|---|---|
| 10 | 0.1 |
| 20 | 0.2 |
| 30 | 0.3 |
| 40 | 0.4 |
- Confirm that \(X\) has a mean of \(\mu = 30\) cubic feet and a standard deviation of \(\sigma = 10\) cubic feet by computing them yourself.
- Derive the sampling distribution for the mean tree volume in a sample of \(n = 2\) trees.
Deriving Probabilities From Discrete Distributions
\(\bf (9)\) Michael Dickson, who plays for the Seattle Seahawks, is a punter in the NFL. The following sampling distribution gives the mean punting distance (in yards) for a sample of \(n=4\) punts. Use this distribution to answer the following questions:
| \(\bar{X} =\) Mean Punt Distance | \(P(\bar{X})\) |
|---|---|
| 35.33 | 0.40 |
| 45.67 | 0.50 |
| 55.00 | 0.08 |
| 60.33 | 0.01 |
| 65.67 | 0.01 |
- What is the probability the mean distance is 55 or more yards?
- What is the probability the mean distance is at least 35.33 yards?
- What is the probability the mean distance is between 35.33 and 45.67 yards?
- What is the probability the mean distance is 35.33 yards or 55 yards?
\(\bf (10)\) A random variable \(X\) follows a binomial distribution with probability of success \(p = 0.43\) and number of trials \(n = 8\). What is the probability of observing fewer than 3 success, that is what is \(P(X < 3)\)?
\(\bf (11)\) Tom Brady is a famous quarterback who recently retired from playing in the National Football League. During his career, the probability that Brady would complete a pass in a given play was \(64.3\%\). If a typical NFL team runs about \(35\) offensive pass plays, what is the probability that Brady would complete exactly \(28\) pass plays in a game?
\(\bf (12)\) A random variable \(W\) follows a Poisson distribution with rate parameter \(\lambda = 1.8\) events per hour. What is the probability of observing exactly four events in a given hour, that is what is \(P(W = 4)\)?
\(\bf (13)\) A factory experiences on average 3 equipment malfunctions per day, following a Poisson distribution with a rate parameter of 3. Calculate the probability that on a randomly selected day, the factory will observe 3 or more equipment malfunctions.
Deriving Probabilities From Continuous Distributions
\({\bf (14)}\) Use the plot below of the probability distribution of a continuous random variable \(X\) to answer the following questions
Use the plot to find the following probabilities:
- What is \(P(X<0.6)\)?
- What is \(P(X>0.3)\)?
- What is \(P(1.5<X<1.8)\)?
\({\bf (15)}\) Use the plot below of the probability distribution of a continuous random variable \(Y\) to answer the following questions
Use the plot to find the following probabilities (recall that the area of a triangle is \(\frac{1}{2} \times \text{base} \times \text{height}\))
- What is \(P(Y>-0.25)\)?
- What is \(P(-0.5<Y<0.25)\)?
\({\bf (16)}\) Use the plot below of the probability distribution of a continuous random variable \(Z\) to answer the following questions
Use the plot to find the following probabilities (recall that the area of a triangle is \(\frac{1}{2} \times \text{base} \times \text{height}\))
- What is \(P(Z<2)\)?
- What is \(P(2<Z<3)\)?
- What is \(P(Z>4)\)?
- Give the approximate value of the \(25th\) percentile of distribution of \(Z\)
\({\bf 17.)}\) Assume the continuous random variable \(X\) follows a normal distribution with mean \(\mu = 25\) and variance \(\sigma^2 = 16\).
Using a \(Z\)-table, find the following (approximate) probabilities for \(X\). You can also use the app in the course website or StatDistributions.com to check your result
- What is \(P(X<15)\)?
- What is \(P(X>28)\)?
- What is \(P(12<X<20)\)?
- What is \(P(X>30)\)?
- What is \(P(25<X<33)\)?
- What is the value of \(X\) marks the \(95th\) percentile of the distribution?
- What is the value of \(X\) marks the \(40th\) percentile of the distribution?
\({\bf 18.)}\) Suppose that a random variable \(X\) follows a normal distribution with mean \(\mu = -5\) and standard deviation \(\sigma = 1.8\). Using the plot of a the \(Z\)-distribution below, shade in the area corresponding to the probability \(P(-2< X <0.5)\)
