Study Guide for Exam 2

This will be a closed-book exam. You will need a calculator. A calculator that performs arithmetic operations, reciprocals, square roots, powers and logarithms (base 10) is sufficient. Graphing calculators are permitted.

To do well on the exam, you should be able to do the following:

Chapter 4

Section 4.2: Random Variables

Denote a random variable by a capital letter and a particular value the random variable can take by a lowercase letter. For example, we might denote the number appearing after the roll of a six-sided die by \(X\) and one of the particular numbers by \(x\).
Explain in what sense a random variable is a number that could have been otherwise.
Explain what a probability distribution tells us about a random variable.
Denote the probability distribution of a random variable \(X\) by \(P(X = x)\), where \(x\) is a particular value the random variable can take.
Define what it means for a random variable to be discrete or continuous, and distinguish between discrete and continuous random variables when given their description.
Construct a probability histogram for a discrete random variable given its probability distribution.
Specify the properties that a probability distribution must satisfy, and check if a given probability distribution satisfies those properties.
Explain what the expectation of a random variable means under a frequency-based model of probability.

Section 4.3: Binomial Probability Distributions

Specify the properties that must be true of a procedure for its outcome to be well-modeled by a binomial random variable.
Identify the roles played by \(n\), \(p\), and \(x\) in the probability distribution of a binomial random variable.
Extract \(n\), \(p\), and \(x\) from the description of a procedure whose outcome is well-modeled by a binomial random variable.
Interpret and use a table like Table A–1 from Appendix A of Triola and Triola to find \(P(X = x; n, p)\).
Use (but do not memorize) the formula \(P(X = x; n, p) = \dfrac{n!}{(n-x)!x!} p^{x} (1 - p)^{n - x}\) to find the probability of \(x\) successes in \(n\) trials.
Convert events such as “at least 1,” “at most 2,” “fewer than 3,” and “more than 10” to the appropriate values of a binomial random variable with \(n\) trials.
Compute the probability of events like those in the previous learning objective using the probability distribution of a binomial random variable.

Section 4.4: Mean, Variance, and Standard Deviation of the Binomial Distribution

Compute the mean, variance, and standard deviation of a binomial random variable with parameters \(n\) and \(p\).
Use the range / standard deviation rule-of-thumb to identify usual and unusual numbers of successes for a binomial random variable with parameters \(n\) and \(p\).
Explain the interpretation of the mean of a binomial random variable under the frequency interpretation of probability.
Construct a probability histogram for a binomial random variable \(X\) given the probability distribution of \(X\).

Chapter 5

Section 5.1: Overview

Compare and contrast density histograms and frequency histograms.
State the properties that must be true of the graph of a function for it to be the graph of a density histogram.
Relate the area under a density histogram between \(x = a\) and \(x = b\) to the probability that a random variable \(X\) following that density histogram falls in the interval \([a, b]\).

Section 5.2: The Standard Normal Distribution

Sketch the graph of the density histogram for a standard normal random variable.
Relate the mean and standard deviation of a density histogram to the mean and standard deviation of a frequency histogram.
Specify the mean and standard deviation of a standard normal random variable.
Use a table like Table A–2 in Triola and Triola to find \(P(Z \leq z)\) for a standard normal random variable \(Z\).
Reason from a sketch of the density histogram of a standard normal random variable to identify \(P(Z \leq z)\), \(P(Z \geq z)\), and \(P(a \leq Z \leq b)\) for a random variable \(Z\).
Use a table like Table A–2 in Triola and Triola to find \(P(Z \leq z)\), \(P(Z \geq z)\), and \(P(a \leq Z \leq b)\) for a standard normal random variable \(Z\).
Use a table like Table A–2 in Triola and Triola to find the \(z\)-values (“\(z\)-scores”) that make the equality \(P(Z \leq z) = p\) true for given value of \(p\) for a standard normal random variable \(Z\).

Section 5.3: Applications of Normal Distributions

Sketch the density histogram for a normal random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\).
Convert a general (nonstandard) normal random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\) to a standard normal random variable \(Z\).
Explain how the conversion \(Z = \dfrac{X - \mu}{\sigma}\) standardizes a nonstandard normal random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\).
Sketch the area under the density histogram of a nonstandard normal random variable \(X\) corresponding to \(P(X \leq x)\), \(P(X \ \geq x)\), and \(P(a \leq X \leq b)\).
For a given value of \(x\), calculate \(P(X \leq x)\) or \(P(X \ \geq x)\) by standardizing \(X\) and using a table like Table A–2 from Triola and Triola.
For given values of \(a\) and \(b\), compute \(P(a \leq X \leq b)\) by standardizing \(X\) and using a table like Table A–2 from Triola and Triola.
Given a probability \(p\), find the value of \(x\) such that \(P(X \leq x) = p\).

Section 5.4: Sampling Distributions and Estimators

Explain why statistics computed from a simple random sample can be modeled using random variables.
Identify the notation \(\bar{X}\) for the sample mean and \(S\) for the sample standard deviation.
Explain the difference between the population distribution modeled by a random variable \(X\) and the sampling distributions of statistics such as the sample mean \(\bar{X}\) or sample standard deviation \(S\).

Section 5.5: The Central Limit Theorem

Given the population mean \(\mu\) and standard deviation \(\sigma\) of a random variable \(X\), find the mean \(\mu_{\bar{X}}\) and standard deviation \(\sigma_{\bar{X}}\) of the sample mean \(\bar{X}_{n}\) of a simple random sample from the population of size \(n\).
Explain, roughly, what the Central Limit Theorem says about the sample mean of a simple random sample from a population.
Identify the rule-of-thumb cutoff for the sample size \(n\) when the normal approximation for a sample mean \(\bar{X}_{n}\) applies.
Compute probabilities such as \(P(\bar{X}_{n} \leq \bar{x})\), \(P(\bar{X}_{n} \geq \bar{x})\), and \(P(a \leq \bar{X}_{n} \leq b)\) given the population mean \(\mu\), standard deviation \(\sigma\), and sample size \(n\) using the normal approximation for the sample mean.
Explain, roughly, why \(X\) from a population and \(\bar{X}_{n}\) from a simple random sample of size \(n\) from the population have different distributions.