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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
- Explain why we need to talk about chance when we consider a a sample from a simple random sample sampling design.
- Define probability in terms of a proportion related to long-run frequencies.
Probability rules
- State the range of values that a probability can take.
- Recognize when a number can and cannot be a probability.
- Relate the probability that an event occurs to the probability that the event does not occur.
- Characterize the probability of an event between 0 and 1 in terms of whether the event will:
- always occur
- never occur
- sometimes occur
Random variables
- 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.”
- Give examples of random variables that result from simple random sampling.
- 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.
- Given the definition of a random variable, determine whether the random variable is discrete or continuous.
Discrete versus continuous probability models
- 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.
- Use a probability distribution to answer probability queries such as P(X>a), P(a<X<b), P(X≤a), etc., for a discrete random variable X.
- State the properties of a density curve for a continuous random variable.
- Relate the probability that a continuous random variable X falls in an interval to its probability curve.
- 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
- Sketch the basic shape of the density curve for a Normal random variable.
- Identify the two parameters that specify the center and spread of the density curve for a Normal random variable.
- Recognize the convention in statistics of using Roman letters for sample statistics (ˉx, s, etc.) and Greek letters for population parameters (μ, σ, etc.).
- Recognize, explain, and use the notation X∼N(μ,σ).
- Sketch a normal density curve for any N(μ,σ) random variable, including its: mean, and mean ± 1, 2, and 3 standard deviations.
The 68–95–99.7 rule
- 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
- State the mean and standard deviation of a standard Normal random variable.
- Explain why, for a Normal random variable with mean μ and standard deviation σ, subtracting its mean and then dividing by its standard deviation transforms the random variable into a standard Normal random variable.
- Compute a Z-score (equivalently, Z-statistic or standardized score) given a data value from a N(μ,σ) random variable.
Finding Normal probabilities
- Convert probability queries of the form
- P(X≤a)
- P(X≥a)
- P(X≤a or X≥b)
- P(a≤X≤b)
into questions about the probability in the left, right, both tails, or body of a Normal density curve, or vice versa.
- Sketch the area corresponding to probabilities in the:
- Left tail
- Right tail
- Both tails
- Body
of a Normal random variable with a given mean and standard deviation.
- Identify which tail
pnorm
returns the probability for, by default.
- Find the corresponding probabilities using
pnorm
in R.
Finding percentiles
- Define the percentile of a distribution.
- Define the quantile of a distribution.
- Convert a question asking for a quantile to a question asking for a percentile or vice versa.
- Compute percentiles or quantiles using
qnorm
.
- Describe what the first letter in the functions
pnorm
and qnorm
are indicating about the function.
- Given a probability p, find the value of a for the equalities p=P(X≤a) and p=P(X≥a).
Chapter 13
Parameters and statistics
- Compare and contrast a parameter of a population and a statistic of a sample.
- Recognize the notation of using Greek letters for parameters and Roman letters for statistics.
- 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
- Define sampling distribution, and relate it to prior concepts in the course such as random variables, probability distributions, and density curves.
- 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 ˉX
- State the mean μˉX of the sample mean ˉX of a simple random sample.
- State the standard deviation σˉX of the sample mean ˉX of a simple random sample.
- Explain, using the mean and standard deviation of the sample mean, why averaging values from a simple random sample is a good idea.
- Relate the sampling distribution of the sample mean to the concepts of accuracy and precision.
- 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
- State under what conditions the sampling distribution of the sample mean is exactly Normal.
- State under what conditions the sampling distribution of the sample mean is approximately Normal.
- 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)
- Distinguish between a descriptive statistic and an inferential statistic, and state what they characterize about a sample or a population.
- Describe the “black box” model of statistical inference.
- State the three main types of inferential statistics we will discuss in this course.
- Define point estimator.
- Distinguish between a point estimator as a procedure and a point estimate as a number.
- 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)
- State the standard error for the sample mean of a simple random sample.
- Explain why the standard error of the sample mean is so-named.
- Determine the probability that the sample mean is within a specified error from the population mean.
- Define a probability c margin of error for a sample mean.
- Compute a probability c margin of error for a sample mean using a critical value from the standard Normal distribution.
- Relate body probabilities c to tail probabilities α.
- Relate the critical values zα 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)
- Explain why the standard error of the sample mean is not useful in practice.
- State and compute an estimate of the standard error of the sample mean.
- Compare and contrast Z-scores and T-scores.
- Explain what a T-score indicates about an observed sample mean.
- Compute the T-score of an observed sample mean given the relevant information about the population and sample.