Terminology
An unknown characteristic about a population
Population Parameter
A type of random variable that can assume only a countable number (finite or infinite) of values such as 0, 1, 2, 3, …
Discrete Random Variable
The distribution that models the number of trials needed for the first success in a series of independent and identical binary outcomes.
Geometric Distribution
For the following question: (a) identify the correct discrete distribution and (b) its parameters, (c) write the probability statement of interest, and (d) write out the appropriate R code to compute the probabilities.
Question: According to a recent national poll, about 40% of Americans believe in the supernatural (e.g., ghosts, werewolves, and vampires). Assuming this percentage is accurate, if 20 people were randomly sampled and asked, “Do you believe in the supernatural?” What is the probability that 11 would say yes?
(a) Binomial Distribution
(b) n = 20 ; p = 0.40
(c) P(X = 11)
(d) dbinom(11, 20, 0.40) or dbinom(11, n, p)
The heights of students in a classroom are normally distributed with a mean of 160 cm and a standard deviation of 7 cm. A student is considered short if their height is less than 150 cm, what is the probability that a randomly selected student is short?
(a) Write out the probability statement in terms of the z-score.
(b) Using the proper distribution for the z-score, draw a normal curve, label the x-axis, and shade the region on the graphic associated with finding the probability of interest.
(c) Using the Z-tables, find and write the associated probability.
(a) z = (150-160)/7 = -1.42; P(Z < -1.42)
(b)
(c) 0.077
A numerical summary of a sample
A type of random variable that can assume values in an interval
Continuous Random Variable
A continuous distribution that is characterized by a constant probability density function, resulting in all outcomes within a specified range having an equal probability of occurring.
Uniform Distribution
For the following question: (a) identify the correct discrete distribution and (b) its parameters, (c) write the probability statement of interest, and (d) write out the appropriate R code to compute the probabilities.
At the beginning of the day, Enterprise has 20 cars in its car lot to rent out to customers. Five (5) automobiles have premium status, and all other cars do not. Once a car is rented out, it is not replaced in the lot. What is the probability that Enterprise rents out less than 3 premium cars from a sample of 10?
(a) Hypergeometric
(b) N = 20; R = 5; n= 10
(c) P(X < 3) = P(X <=2)
(d) phyper(2, R, N-R, n)
A certain species of fish in a pond has a length that follows a normal distribution with a mean of 20 cm and a standard deviation of 3 cm. If a fish is considered a perfect catch if its length is between 18 cm and 24 cm, what is the probability that a randomly selected fish is a perfect catch?
(a) Write out the probability statement.
(b) Draw a normal curve, Label the x-axis, and shade the region on the graphic associated with finding the probability of interest.
(c) Using the R commands, what would you type to find the associated probability?
(a) P(18 < X < 24)
(b)
(c) pnorm(24, mean = 20, sd = 3) - pnorm(18, mean = 20, sd = 3)
The distribution applies when an experimental trial has only two possible outcomes (success/failure), the trials are independent, the probability of success is the same for each trial, and a fixed number of trials is considered.
Binomial Distribution
A continuous distribution that is symmetric, bell-shaped, and characterized by its mean and standard deviation, with most observations clustering around the mean.
Normal Distribution
A guideline for understanding the spread of data in a normal distribution
Empirical Rule (or 68-95-99.7)
For the following question: (a) identify the correct discrete distribution and (b) its parameters, (c) write the probability statement of interest, and (d) write out the appropriate R code to compute the probabilities.
Question: In a video game, a player attempts to defeat a boss monster known to drop a rare item with a probability of 0.1 on each attempt. What is the probability that the player defeats the boss monster for the first time on their fifth attempt?
(a) Geometric
(b) p =0.1
(c) P(X = 5)
(d) dgeom(5-1,p)
A certain species of fish in a pond has a length that follows a normal distribution with a mean of 20 cm and a standard deviation of 3 cm. If a fish is considered a perfect catch if its length is between 18 cm and 24 cm, what is the probability that the average of 9 fish is between the range of perfect catch?
(a) Write out the probability statement.
(b) Draw a normal curve, Label the x-axis, and shade the region on the graphic associated with finding the probability of interest.
(c) Using the R commands, what would you type to find the associated probability?
(a) P(18 < X-bar < 24)
(b)
(c) pnorm(24, mean = 20, sd = 3/sqrt(9)) - pnorm(18, mean = 20, sd = 3/sqrt(9))
The distribution that characterizes the number of events that occur within a fixed interval of time or space.
Poisson Distribution
The distribution describes the number of successes drawn without replacement from a finite population of distinct elements, where the population consists of both successes and failures.
Hypergeometric Distribution
A method for estimating the parameters of a statistical model by maximizing the likelihood function.
Maximum Likelihood Estimation
For the following question: (a) identify the correct discrete distribution and (b) its parameters, (c) write the probability statement of interest, and (d) write out the appropriate R code to compute the probabilities.
Question: On average, ten (10) students enrolled at Virginia Tech have their automobiles ticketed on campus during the week. What is the probability that more than eight (8) students have their automobiles ticketed during the week?
(a) Poisson
(b) lambda = 10
(c) P(X > 8) = P(X >=9) = 1 - P(X <=8)
(d) 1 - ppois(8, lambda)
The Nature’s Bakery Fig Bar Company creates two ideal sizes of fig bars. The company has collected historical data and calculated the average fig bar length as 11 centimeters and a standard deviation of 5. What is the probability that a Fig Bar is greater than 17 centimeters?
(a) Write out the probability statement.
(b) Draw a normal curve, Label the x-axis, and shade the region on the graphic associated with finding the probability of interest.
(c) Using the R commands, what would you type to find the associated probability?
(a) P(X > 17)
(b)
(c)
pnorm(17, mean = 11, sd = 5, lower.tail = FALSE)
or
1 - pnorm(17, mean = 11, sd = 5)
A statistical function applied to sample data that produces a single numerical value, known as an estimate, which serves as the best guess for an unknown population parameter.
Point Estimator
The specific numerical value generated by a point estimator based on sample data.
Point Estimate
A probability distribution of a statistic (such as the mean, variance, etc.) calculated from multiple samples of the same sample size taken from the same population.
Sampling Distribution
For the following question: (a) identify the correct discrete distribution and (b) its parameters, (c) write the probability statement of interest, and (d) write out the appropriate R code to compute the probabilities.
Question: In a basketball tournament, a team must win 3 games to qualify for the finals. The team has a 60% chance of winning each game. What is the probability that the team will win the third game exactly on their 5th attempt?
(a) Negative Binomial
(b) r = 3; p = 0.60
(c) P(X = 5)
(d) dnbinom(5-r, r, p)
The Nature’s Bakery Fig Bar Company creates two ideal sizes of fig bars. The company has collected historical data and calculated the average fig bar length as 11 centimeters and a standard deviation of 5.
A fig bar is considered an “ideal” size if it measures 10 – 10.5 centimeters or 11.5 – 12 centimeters. What is the probability that a sample mean of 100 fig bars is in the ideal ranges?
(a) Write out the probability statement.
(b) Draw a normal curve, Label the x-axis, and shade the region on the graphic associated with finding the probability of interest.
(c) Using the R commands, what would you type to find the associated probability?
(a) P( 10 < x-bar <10.5) + P(11.5 < x-bar <12)
(b)
(c) Mean = 11; standard deviation = 5/sqrt(100) = 0.5;
[pnorm(10.5, mean, std) - pnorm(10, mean, std)]+ [pnorm(12, mean, std) - pnorm(11.5, mean, std)]