Leaving Certificate - Probability and Statistics Higher Level

Learning Outcomes

Students will learn about:1.1 Counting

• - count the arrangements of n distinct objects (n!)
• - count the number of ways of arranging r objects from n distinct objects
• - count the number of ways of selecting r objects from n distinct objects

1.2 Concepts of probability

• - discuss basic rules of probability (AND/OR, mutually exclusive) through the use of Venn diagrams
• - calculate expected value and understand that this does not need to be one of the outcomes
• - recognise the role of expected value in decision making and explore the issue of fair games
• - extend their understanding of the basic rules of probability (AND/OR, mutually exclusive) through the use of formulae
• - use the Addition Rule, Multiplication Rule (Independent events), Multiplication Rule (General case)
• - solve problems involving conditional probability in a systematic way

1.3 Outcomes of random processes

• - find the probability that two independent events both occur
• - apply an understanding of Bernoulli trials
• - solve problems involving up to 3 Bernoulli trials
• - calculate the probability that the 1st success occurs on the nth Bernoulli trial where n is specified
• - solve problems involving calculating the probability of k successes in n repeated Bernoulli trials (normal approximation not required)
• - calculate the probability that the kth success occurs on the nth Bernoulli trial
• - use simulations to explore the variability of sample statistics from a known population and to construct sampling distributions
• - solve problems involving reading probabilities from the normal distribution tables

1.4 Statistical reasoning with an aim to becoming a statistically aware consumer

• - work with different types of bivariate data

1.5 Finding, collecting and organising data

• - discuss different types of studies: sample surveys, observational studies and designed experiments
• - design a plan and collect data on the basis of above knowledge
• - recognise the importance of randomisation and the role of the control group in studies
• - recognise biases, limitations and ethical issues of each type of study
• - select a sample (stratified, cluster, quota – no formulae required, just definitions of these)
• - design a plan and collect data on the basis of above knowledge

1.6 Representing data graphically and numerically1.6a Graphical

• - describe the sample (both univariate and bivariate data) by selecting appropriate graphical or numerical methods
• - explore the distribution of data, including concepts of symmetry and skewness
• - compare data sets using appropriate displays, including back-to-back stem and leaf plots
• - determine the relationship between variables using scatterplots
• - recognise that correlation is a value from -1 to +1 and that it measures the extent of the linear relationship between two variables
• - match correlation coefficient values to appropriate scatter plots
• - understand that correlation does not imply causality
• - analyse plots of the data to explain differences in measures of centre and spread
• - draw the line of best fit by eye
• - make predictions based on the line of best fit
• - calculate the correlation coefficient by calculator

1.6b Numerical

• - recognise standard deviation and interquartile range as measures of variability
• - use a calculator to calculate standard deviation
• - find quartiles and the inter-quartile range
• - use the interquartile range appropriately when analysing data
• - recognise the existence of outliers
• - recognise the effect of outliers
• - use percentiles to assign relative standing

1.7 Analysing, interpreting and drawing inferences from data

• - interpret a histogram in terms of distribution of data
• - make decisions based on the empirical rule
• - recognise the concept of a hypothesis test
• - calculate the margin of error for a population proportion
• - conduct a hypothesis test on a population proportion using the margin of error

1.8 Synthesis and problem-solving skills

• - explore patterns and formulate conjectures
• - explain findings
• - justify conclusions
• - communicate mathematics verbally and in written form
• - apply their knowledge and skills to solve problems in familiar and unfamiliar contexts
• - analyse information presented verbally and translate it into mathematical form
• - devise, select and use appropriate mathematical models, formulae or techniques to process information and to draw relevant conclusions