Leaving Certificate - Probability and Statistics Higher Level
Learn to make informed decisions with this Probability and Statistics for the Irish Leaving Certificate course.
Description
This course begins with an introduction to data. You will study the different types of data including categorical data, numerical data, and continuous and discrete data. You will also look into the frequency table, pie chart, histogram, and bar chart. This course will also discuss the methods of representing data, the measures of central tendency and central spread, stem-and-leaf plots, skewness, and correlation as well as the mean, median, mode, range, and interquartile range.
You will then look into probability which is concerned with the chance that something may happen. You will learn how to calculate for the probability of events occurring using rules or diagrams such as tables, tree, and Venn diagrams. You will learn about the expected value which is used widely in insurance industries and casino games, study binomial and normal distribution, hypothesis testing which is a statistical method of proving/disproving a hypothesis, and more.
Probability is concerned with the likelihood of an event happening and a combination of probability and statistics can be used to prove/disprove a given conjecture or statement. Upon the completion of this course, you will gain better insight into the mathematical aspects of probability and statistics with real-life applications to help you make informed decisions. Sign up for this course and gain a better understanding of Probability and Statistics today!
Modules
Module 13: Probability
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Fundamental Principle of Counting
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Calculating the outcome
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Permutations and combinations
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Introduction to probability
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Probabilities
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Finding probabilities theoretically
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Basic rules of probability
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Venn diagrams
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Conditional probability
Module 15: Binomial Distribution
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Binomial probability function and distribution
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The number of successes in a given number of trials
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Binomial distribution: Bernoulli trials
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Bernoulli extended
Module 16: Normal Distribution
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Introduction to the normal distribution
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The normal curve
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Continuous random variables and the normal distribution
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Calculation of probabilities for a normal distribution
Module 17: Hypothesis Testing
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Hypothesis testing
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Hypothesis testing - Example
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Hypothesis testing - Formal Procedure
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Worked example
Module 1: Types of Data and Sampling
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Introduction to data
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Overview of data
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Types of data
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Categorical data
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Numerical data
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Continuous and discrete data
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Sample types
Module 3: Methods of Representing Data
Module 6: Stem-and-Leaf Plots
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Introduction to stem-and-leaf
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Stem-and-leaf plots
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Back to back stem plots
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Stem-and-leaf diagrams 1
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Stem-and-leaf diagrams 2
Module 5: Histograms and Bar Charts
Module 8: Scatter Plots - Line of Best Fit
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Scatterplots
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Strength of association
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Introduction to the regression line
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Finding the equation of a regression line
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Intrepretation of slope and intercept
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Practice question
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Scatter plots and linear models
Module 9: Correlation
Module 12: Analyse Data
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Mean from frequency tables - Discrete data
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Frequency and graphs - Overview
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Summarising data - Overview
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Mode
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Mean
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Median
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Mode, mean, median
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Comparing mode, mean and median
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Range of data
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Inter-quartile range
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Review - Summarising data
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Standard deviation and normal distribution
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Standard deviation and calculator
Module 4: Pie Charts
Module 10: Measures of Central Tendency
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Measures of central tendency: Mean, mode and median
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Mode
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Mean
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Median
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Mode, mean and median
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Comparing mode, mean and median
Module 11: Measures of Central Spread
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Measures of central spread
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Range of data
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Inter-quartile range
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Standard deviation as a measure of spread
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Summarising data - Overview
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Soccer activity
Module 14: Expected Value
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Probability and relative frequency
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Short-run coin tossing
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Short-run dice rolling
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Predicting from past experience
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Towards probability with coins
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Towards probability with dice
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Probability as long-run relative frequency
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Mean and variance of a discrete random variable
Module 2: Frequency Tables
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Frequency and graphs - Overview
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Frequency tables with nominal data
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Frequency tables with discrete data
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Frequency tables - Discrete data and summary statistics
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Mean from frequency tables - Discrete data
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Interpreting bar graphs
Module 7: Skewness
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Comparing mean, mode and median
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Symmetry and skew of a distribution
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Negative skewness – Left skewness
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Positive skewness – Right skewness
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Probability intervals
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Comparing sample and population
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Probability intervals - Examples
Module 18: Problem Solving
End of Course Information
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
Certification
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