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Summary - Probability and Statistics ENG
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statisticsBenoît Waûters
,Inhoud
1. Statistics + Data and Statistical Thinking ......................................................................................... 4
1.1. Types of statistical applications............................................................................................... 4
1.2. Fundamental Elements of Statistics ........................................................................................ 4
1.3. Types of Data ........................................................................................................................... 4
1.4. Collecting data ......................................................................................................................... 4
1.4.1. Data collection methods ................................................................................................. 4
1.4.2. Sorts of Samples .............................................................................................................. 5
1.4.3. Sorts of results ................................................................................................................. 5
2. Methods for Describing Sets of Data............................................................................................... 6
2.1. Describing qualitative data ...................................................................................................... 6
2.2. Graphical methods for describing Quantitative Data ............................................................. 6
2.3. Numerical measures of central tendency ............................................................................... 6
2.4. Numerical measures of variability ........................................................................................... 7
2.5. Interpreting the Standard Deviation ....................................................................................... 7
2.6. Numerical measures of relative standing................................................................................ 8
2.7. Methods for detecting outliers: box plots & z-scores ............................................................. 8
2.8. Graphing bivariate relationships ............................................................................................. 9
2.9. Distorting the truth with descriptive statistics ........................................................................ 9
3. Probability ..................................................................................................................................... 11
3.1. Events, sample spaces & probability ..................................................................................... 11
3.2. Unions and Intersections....................................................................................................... 11
3.3. Complementary Events ......................................................................................................... 11
3.4. The additive rule and mutually exclusive events .................................................................. 12
3.5. Conditional probability .......................................................................................................... 12
3.6. The multiplicative rule and independent events .................................................................. 12
3.7. Counting Rules....................................................................................................................... 12
3.8. Bayes’ rule ............................................................................................................................. 13
4. Discrete random variables............................................................................................................. 14
4.1. Discrete vs. continuous variables .......................................................................................... 14
4.2. Probability distributions for discrete random variables........................................................ 14
4.3. Expected values of discrete random variables ...................................................................... 14
4.4. The binomial random variable .............................................................................................. 15
4.5. The Poisson random variable ................................................................................................ 15
5. Continuous random variables ....................................................................................................... 17
5.1. Continuous probability distributions..................................................................................... 17
, 5.2. The uniform distribution ....................................................................................................... 17
5.3. The normal distribution......................................................................................................... 17
5.4. Descriptive methods for assessing normality ....................................................................... 18
5.5. Approximating a binomial distribution with a normal distribution ...................................... 18
5.6. Bivariate gaussian distributions ............................................................................................ 19
5.7. Linear combinations of random variables ............................................................................. 19
6. Sampling distributions ................................................................................................................... 20
6.1. The concept of a sampling distribution ................................................................................. 20
6.2. The sampling distribution of 𝑥 and the central limit theorem .............................................. 20
7. Inferences based on a single sample Estimations with confidence intervals ..................................................... 22
7.1. Identifying and estimating the target parameter ................................................................. 22
7.2. Confidence Interval for a Population Mean: Normal (z) Statistics n > 30 ............................. 22
7.3. Confidence interval for a population mean: t-Statistic ......................................................... 23
7.4. Large sample confidence interval for a population proportion ............................................ 24
7.5. Determining the sample size ................................................................................................. 24
8. Inferences based on a single sample ............................................................................................. 26
8.1. The elements of a test of hypothesis .................................................................................... 26
8.2. Formulating hypotheses and setting up the rejection region ............................................... 27
8.3. Test of hypothesis about a population mean: normal (𝑧) statistic ...................................... 27
8.4. Observed significance levels: 𝑝 − 𝑣𝑎𝑙𝑢𝑒𝑠 ............................................................................ 28
8.5. Test of hypothesis about a population mean= student’s 𝑡 − statistic .................................. 29
8.6. Large-sample test of hypothesis about a population proportion ......................................... 29
8.7. Calculating 𝑇𝑦𝑝𝑒 𝐼𝐼 𝐸𝑟𝑟𝑜𝑟 probabilities: more on ‘𝛽’ .......................................................... 29
9. Inferences based on 2 samples ..................................................................................................... 32
9.1. Identifying the target parameter .......................................................................................... 32
9.2. Comparing 2 population means: independent sampling ...................................................... 32
9.3. Comparing 2 population means: independent sampling small sample ................................ 33
9.4. Comparing 2 population means: paired difference experiments ......................................... 34
9.5. Comparing two population proportions: independent sampling ......................................... 35
9.6. Determining the sample size ................................................................................................. 35
9.7. Comparing two population variances: independent sampling ............................................. 36
10. Simple linear regression ............................................................................................................ 38
10.1. Probabilistic models .......................................................................................................... 38
10.2. Fitting the model: the least squares approach.................................................................. 38
10.3. Model assumptions ........................................................................................................... 39
10.4. Assessing the utility of the model: making inferences about the slope 𝛽1 ...................... 40
, 10.5. The coefficients of correlation and determination ........................................................... 41
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