Math 445 – Mathematical Statistics

Review Sheet for Exam 1

The first exam is in class on Wednesday, February 10. It will cover the textbook material in Chapters 1, 6, 7 (no Section 7.3 nor 7.4), 8 (no Section 8.4), and 9 (no Section 9.3 nor Section 9.5, except for the little bit about practical significance), as well as any supplementary material/details I provided in lecture. You do not need to memorize the probability density functions of the common distributions. (If I ask a question about say, the exponential distribution, I will provide the density, and mean and variance, if necessary.) You do need to know how to calculate means and variances for discrete and continuous random variables, properties of expectation and variance (e.g., what happens when multiplying by a constant, or when summing or multiplying independent variables?), and how to use p.d.f.s (p.m.f.s) to find probabilities (and c.d.f.s).

Be sure you understand all the homework problems and examples we did in class (if you don’t, please see me in office hours to talk about your questions). Furthermore, for the exam you must know the definitions (conceptual and formulaic), terminologies, and notations in your bones (that is, there won’t be time to leisurely recall definitions during the exam). You must actively practice for the exam. That is, do not simply read through your solutions to lecture examples and homework problems. Actually re-work through the problems. (You can do additional textbook problems for extra practice—some answers are included in the back of the book, and you can talk with me if you have questions.)

Specific Topics

• Numerical and graphical summaries for investigating one variable or the relationship between variables (use of and interpretation of these—not the actual calculation)

• Central Limit Theorem: Sampling distribution of the sample mean and sample total (in both binomial setting and non-binomial setting)—including the values of the mean and standard deviation for the sampling distribution; Sample size checks to ensure n is “large” and the CLT applies (that is, that it will provide a good approximation); use of the continuity correction when using the normal distribution to approximate binomial probabilities

• Distribution of a linear combination of normal random variables; Normal distribution in general (standardization and using Table A.3 to find normal areas and percentiles)

• Chi-squared distribution
• using Table A.7 to find areas and percentiles
• the square of a standard normal random variable is a Chi-squared random variable with 1 degree of freedom
• the sum of independent Chi-squared random variables is also chi-squared (with degrees of freedom equal to the sum of the individual degrees of freedom)

• Definitions of and properties of and (both are unbiased); When sampling from a distribution is , and is Chi-squared with (n – 1) degrees of freedom—furthermore, and are independent

• Definition of a t random variable and using the definition to show that a random variable has a t distribution; Using Table A.5 to find areas and percentiles; (For this test, you don’t need to know the definition of an F random variable)

• Properties of estimators: mean-square error and unbiasedness/biasedness

• Parameter estimation methods: method of moments and method of maximum likelihood; Invariance property of maximum-likelihood estimators

• General idea and derivation of a confidence interval; Careful interpretation of a confidence interval (what exactly is meant by “confidence”)

• Confidence interval for a population mean (both when is known and unknown); Prediction interval for single value; Sample size determination (based on a desired confidence level and margin of error)

• Confidence interval for a population proportion—the difference between the score and Wald intervals (why the score interval is better and how they are essentially the same for large n); If needed, the formula for the score interval will be provided on the exam

• Bootstrap confidence intervals (the general idea—of course you won’t actually do any bootstrapping during the exam, but you may need to answer general questions about the bootstrap idea)

• Significance testing: i) statement of hypotheses, ii) check of conditions, iii) calculation of test statistic, iv) calculation of p-value, and v) interpretation of the results in the context of the problem, including a check for practical significance

• Definitions of Type I error, Type II error, and power

• Calculating , and power (only for a significance test of a population mean when is known)—not using formulas in book, but working through all the details (same for sample-size determination in next bullet point)

• Sample size determination for a test (based on a desired power, significance level, and a mean value in the alternative)—only for a one-sided test of a population mean when is known

• Relationship between confidence intervals and significance testing; Practical significance