| Margaret E. M. Thomas | Department of Mathematics and Statistics |
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. (This course is NOT on Avenue to Learn. Please do not go looking for it there or ask me why you cannot find it there!)..For news about the course, it is your responsibility to check the Course Website regularly.
Summaries of lectures and notes from class will be posted here as the semester progresses. Lecture notes will typically be posted all together once a week, but may be posted earlier if certain topics relate to an assignment with an earlier deadline.
The lecture notes below are typically augmented versions of what was presented on screen during class; the augmentations are principally to be found in the form of extra comments (and corrections) written in light blue. Please send me an email if you ever find any mistakes or there are glitches in the documents.
You are strongly advised to go through these notes alongside your own notes carefully to make sure that you understand everything that was discussed!
(Tues 8 January 2019) -- Introduction to course, including introduction to the Course Website . Introduction to the role of Statistics and Probability Theory. 2-1 Sample spaces. (Wed 9 January 2019) -- More on 2-1: Events (distributive laws, De Morgan's laws), start of counting techniques (multiplication rule). (Fri 11 January 2019) -- More on 2-1: Further counting techniques (k-permutations, permutations of similar objects, r-combinations). (Tues 15 January 2019) -- 2-2: Definition of probability, randomness, axioms of probability; 2-3: Addition Rule for probability. (Wed 16 January 2019) -- More on 2-3: Addition Rule for probability; 2-4: Conditional probability; 2-5: More on conditional probability, Total Probability Rule. (Fri 18 January 2019) -- More on 2-5: More on conditional probability; 2-6: Independence of events. (Thanks to Dr Hoppe for these notes.) (Tues 22 January 2019) -- 2-8, 3-1: Random variables (in particular discrete random variables); 3-2: Probability mass functions (pmf); 3-3 Cumulative distribution functions (cdf). (Wed 23 January 2019) -- More on 3-2, 3-3 (Probability mass functions and cumulative distribution functions); 3-4 Mean and Variance of a discrete random variable. (Fri 25 January 2019) -- More on 3-4 Expected Value of a function of a discrete random variable; 3-6: Binomial distribution (definition, mean and variance). (Tues 29 January 2019) -- 3-7: Geometric and Negative Binomial distributions; 3-8: Hypergeometric distribution. (The slides that Dr McNicholas used during the lecture can be found by logging on to the Course Website ). (Wed 23 January 2019) -- 3-9: Poisson distribution. *** UP TO THIS POINT IS THE SYLLABUS FOR MIDTERM TEST 1. *** 4-1: Continuous random variables. (Fri 1 February 2019) -- 4-2 Probability density functions (pdf); 4-3: Cumulative distribution functions; 4-4: Mean and variance of a continuous random variable (definition). (Tues 5 February 2019) -- More on 4-4: Mean and variance of a continuous random variable; 4-6: Normal distribution. (Fri 8 February 2019) -- More on 4-6: Normal distribution; 4-7: Normal approximation to the binomial distribution. (Wed 13 February 2019) -- 4-8: Exponential distribution; 5-1 Two random variables (5-1.1 joint probability distribution functions). (Fri 15 February 2019) -- More on 5-1 Two random variables (5-1.2 marginal distribution functions, 5-1.4 independence of random variables); 5-2 Covariance and correlation of two random variables; 5-4 linear functions of random variables. Here is a document full of worked examples for the topics covered in Lecture 16 that you should work through independently. Please note the Important Fact about linear combinations of Normally distributed random variables on pp. 7-8 which I neglected to mention in class. (Tues 26 February 2019) -- 6-1 Numerical Summaries of Data (sample mean, sample median, sample mode, sample variance, sample standard deviation, sample range); 6-2: Stem and Leaf Plots. (Wed 27 February 2019) -- More on 6-2: Stem and Leaf Plots (quartiles and percentiles); 6-3: Frequency distributions and histograms; 6-4: Box (and whisker) plots. (Fri 1 March 2019) -- More on 6-4: Box (and whisker) plots; 6.7: Probability plots; 7-1, 7-2: Point Estimation and Sampling Distributions (parameters, random samples, statistics, estimators, estimates). (Tues 5 March 2019) -- More on 7-1: Point Estimation; 7-3: Concepts of Point Estimation (bias, trimmed mean, MVUE (Minimum Variance Unbiased Estimator), standard error, estimated standard error, mean square error, relative efficiency, optimal estimator). (Wed 6 March 2019) -- More on 7-2: Central Limit Theorem; 8-1: Confidence Intervals (general definition of a $100(1-\alpha)\%$ confidence interval; confidence interval for the mean of a Normal distribution, variance known). (Fri 8 March 2019) -- More on 8-1: Confidence Intervals (confidence interval for the mean of a Normal distribution, variance known -- margin of error (ME), one-sided confidence bounds; confidence interval for the mean of any distribution, variance known, large sample size; confidence interval for the mean of any distribution, variance unknown, large sample size); 8-2: Confidence interval for the mean of a Normal distribution, variance unknown -- (Student) t-distribution, t-tables). (Tues 12 March 2019) -- More on 8-2: Confidence interval for the mean of a Normal distribution, variance unknown ((student) t-distribution, t-tables); 8-4: Confidence interval for the population proportion (large sample). (Wed 13 March 2019) -- 9-1: Hypothesis testing (null hypothesis $H_0$, alternative hypothesis $H_1$, test statistic, critical values/regions, Type I and Type II errors, significance level $\alpha$, power $1-\beta$). (Notes corrected 19/03/2019.) (Fri 15 March 2019) -- More on 9-1: Hypothesis testing (p-value, relationship to confidence intervals); 9-2: Hypothesis testing for the mean of a Normal distribution, variance known ($Z_0$, $z$-test). (Tues 19 March 2019) -- More on 9-2: Hypothesis testing for the mean of a Normal distribution, variance known (more on $z$-test, Type II Error $\beta$ and sample size $n$). (Wed 20 March 2019) -- 9-3: Hypothesis testing for the mean of a Normal distribution, variance unknown ($T_0$, $t$-test); 9-5: Hypothesis testing for the population proportion (test statistic $Z_0$, critical regions and p-values). (Fri 22 March 2019) -- More on 9-5: Hypothesis testing for the population proportion (relationship between Type II Error $\beta$ and sample size $n$); 10-2: Hypothesis testing and confidence intervals for the difference of means of two normal populations, variances unknown (equal and not equal cases). (Tues 26 March 2019) -- More on 10-2: Hypothesis testing and confidence intervals for the difference of means of two normal populations, variances unknown (examples); 11-1 and 11-2: Simple Linear Regression (residuals, least squares regression line, least squares estimates/estimators $\hat{\beta}_0, \hat{\beta}_1$). (Wed 27 March 2019) -- More on 11-1 and 11-2: Simple Linear Regression (variance of the error, sums of squares error $SS_E$, total sum of squares $SS_T$, regression sum of squares $SS_R$); 11-3: Properties of the Least Squares Estimators (mean, variance and estimated standard error of $\hat{\beta}_0, \hat{\beta}_1$). (Fri 29 March 2019) -- 11-4: Hypothesis Tests in Simple Linear Regression (t-test for $\hat{\beta}_1$); 11-5 Confidence intervals (on $\hat{\beta}_1$ and the expected value of $Y$ at $X=x_0$); 11-6: Prediction of New Observations (prediction intervals for the future value of $Y$ at $X=x_0$). (Tues 2 April 2019) -- More on 11-4: Hypothesis Tests in Simple Linear Regression (Analysis of Variance Approach (ANOVA) and $F$-tests); 11-7: Adequacy of the Regression Model (probability plots for residuals, coefficient of determination $R^2$); 11-8 Correlation ($R$, and $R$ v. $R^2$). (Wed 3 April 2019) -- 13-2: Single-factor Experiments and Analysis of Variance (ANOVA) (factors, treatments, completely randomized experimental design, Experiment-wise Error Rate EER, dot notation). (Fri 5 April 2019) -- More on 13-2: Single-factor Experiments and Analysis of Variance (ANOVA) (total sums of squares $SS_T$, treatment sums of squares $SS_{\textrm{Treatments}}$, error sums of squares $SS_E$, mean squares treatments $MS_{\textrm{Treatments}}$, mean squares error $MS_E$, $F$-test on equal means/no treatment effects). (Tues 9 April 2019) -- More on 13-2: Single-factor Experiments and Analysis of Variance (ANOVA) (confidence intervals using ANOVA for single treatment means and the difference in treatment means, tests on pairs of treatment means; testing assumptions with ANOVA; unbalanced experiments). (Wed 17 April 2019) -- Review of "End of Course"/"Part 3" material, in particular focussing on Confidence Intervals and Hypothesis Tests (Chapters 8, 9 and 10).