Study Guide for Exam 2

This will be a closed-book exam. You will be allowed to use R on a school computer for computations. This means you will not have access to, nor will you need to use, anything beyond Base R (e.g. mosaic, MUsaic, etc.) for the exam.

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

Chapter 9

The idea of probability

  1. Explain why we need to talk about chance when we consider a a sample from a simple random sample sampling design.
  2. Define probability in terms of a proportion related to long-run frequencies.

Probability rules

  1. State the range of values that a probability can take.
  2. Recognize when a number can and cannot be a probability.
  3. Relate the probability that an event occurs to the probability that the event does not occur.
  4. Characterize the probability of an event between 0 and 1 in terms of whether the event will:

Random variables

  1. Explain why some variables are considered “random” in statistics, and explain in what sense a random variable is a “number that could have been otherwise.”
  2. Give examples of random variables that result from simple random sampling.
  3. Recognize and use the notation of an upper case letter (like \(X, Y,\) or \(Z\)) for a random variable, and a lower case letter (like \(x, y,\) or \(z\)) for a particular value the random variable could take.
  4. Given the definition of a random variable, determine whether the random variable is discrete or continuous.

Discrete versus continuous probability models

  1. State the properties of a probability distribution for a discrete random variable, and identify whether a given table can or cannot be a probability distribution.
  2. Use a probability distribution to answer probability queries such as \(P(X > a)\), \(P(a < X < b)\), \(P(X \leq a)\), etc., for a discrete random variable \(X\).
  3. State the properties of a density curve for a continuous random variable.
  4. Relate the probability that a continuous random variable \(X\) falls in an interval to its probability curve.
  5. Given a rectangular or triangular density curve for a random variable \(X\), find the probability that \(X\) falls in a given interval.

Chapter 11

Normal distributions

  1. Sketch the basic shape of the density curve for a Normal random variable.
  2. Identify the two parameters that specify the center and spread of the density curve for a Normal random variable.
  3. Recognize the convention in statistics of using Roman letters for sample statistics (\(\bar{x}\), \(s\), etc.) and Greek letters for population parameters (\(\mu\), \(\sigma\), etc.).
  4. Recognize, explain, and use the notation \(X \sim N(\mu, \sigma)\).
  5. Sketch a normal density curve for any \(N(\mu, \sigma)\) random variable, including its: mean, and mean \(\pm\) 1, 2, and 3 standard deviations.

The 68–95–99.7 rule

  1. Use the 68–95–99.7 rule to reason about probabilities in the body of the density curve of a Normal random variable.

The standard Normal distribution

  1. State the mean and standard deviation of a standard Normal random variable.
  2. Explain why, for a Normal random variable with mean \(\mu\) and standard deviation \(\sigma\), subtracting its mean and then dividing by its standard deviation transforms the random variable into a standard Normal random variable.
  3. Compute a \(Z\)-score (equivalently, \(Z\)-statistic or standardized score) given a data value from a \(N(\mu, \sigma)\) random variable.

Finding Normal probabilities

  1. Convert probability queries of the form
    into questions about the probability in the left, right, both tails, or body of a Normal density curve, or vice versa.
  2. Sketch the area corresponding to probabilities in the:
    of a Normal random variable with a given mean and standard deviation.
  3. Identify which tail pnorm returns the probability for, by default.
  4. Find the corresponding probabilities using pnorm in R.

Finding percentiles

  1. Define the percentile of a distribution.
  2. Define the quantile of a distribution.
  3. Convert a question asking for a quantile to a question asking for a percentile or vice versa.
  4. Compute percentiles or quantiles using qnorm.
  5. Describe what the first letter in the functions pnorm and qnorm are indicating about the function.
  6. Given a probability \(p\), find the value of \(a\) for the equalities \(p = P(X \leq a)\) and \(p = P(X \geq a)\).

Chapter 13

Parameters and statistics

  1. Compare and contrast a parameter of a population and a statistic of a sample.
  2. Recognize the notation of using Greek letters for parameters and Roman letters for statistics.
  3. Given the description of a statistical problem, identify whether a numerical characteristic is a population parameter or a sample statistic.

Statistical estimation and sampling distributions

  1. Define sampling distribution, and relate it to prior concepts in the course such as random variables, probability distributions, and density curves.
  2. Explain why we can model the statistic of a sample as a random variable when the sample was a simple random sample from a population.

The sampling distribution of \(\bar{X}\)

  1. State the mean \(\mu_{\bar{X}}\) of the sample mean \(\bar{X}\) of a simple random sample.
  2. State the standard deviation \(\sigma_{\bar{X}}\) of the sample mean \(\bar{X}\) of a simple random sample.
  3. Explain, using the mean and standard deviation of the sample mean, why averaging values from a simple random sample is a good idea.
  4. Relate the sampling distribution of the sample mean to the concepts of accuracy and precision.
  5. Given a population mean and standard deviation, compute the mean and standard deviation of the sample mean from a simple random sample of the population.

The central limit theorem

  1. State under what conditions the sampling distribution of the sample mean is exactly Normal.
  2. State under what conditions the sampling distribution of the sample mean is approximately Normal.
  3. Answer probability queries about the sample mean from a simple random sample given the relevant characteristics of the sample and the population.

Chapter 14

Statistical estimation (and Lecture 11 Lecture Notes)

  1. Distinguish between a descriptive statistic and an inferential statistic, and state what they characterize about a sample or a population.
  2. Describe the “black box” model of statistical inference.
  3. State the three main types of inferential statistics we will discuss in this course.
  4. Define point estimator.
  5. Distinguish between a point estimator as a procedure and a point estimate as a number.
  6. Discuss the main properties of the sample mean of a random sample from a population as a point estimator for the population’s mean.

Margin of error and confidence level (and Lecture 11 Lecture Notes)

  1. State the standard error for the sample mean of a simple random sample.
  2. Explain why the standard error of the sample mean is so-named.
  3. Determine the probability that the sample mean is within a specified error from the population mean.
  4. Define a probability \(c\) margin of error for a sample mean.
  5. Compute a probability \(c\) margin of error for a sample mean using a critical value from the standard Normal distribution.
  6. Relate body probabilities \(c\) to tail probabilities \(\alpha\).
  7. Relate the critical values \(z_{\alpha}\) of a standard Normal distribution to its quantiles, and compute such a critical value using R.

Chapter 17

The \(t\) distributions (and Lecture 11 Lecture Notes)

  1. Explain why the standard error of the sample mean is not useful in practice.
  2. State and compute an estimate of the standard error of the sample mean.
  3. Compare and contrast \(Z\)-scores and \(T\)-scores.
  4. Explain what a \(T\)-score indicates about an observed sample mean.
  5. Compute the \(T\)-score of an observed sample mean given the relevant information about the population and sample.