Advanced Statistical analysis – Random variables

This lesson comprises two (2) master classes focusing on:

• Continuous random variables
• Probability density function
• Cumulative distribution function
• Normally distributed random variables
• z-scores
• Empirical rule
• Using z-scores to compare data sets
• Using z-scores to calculate probabilities

Content:

MA-S3.1

• Use relative frequencies and histograms obtained from data to estimate probabilities associated with a continuous random variable
• Understand and use the concepts of a probability density function of a continuous random variable
  • know the two properties of a probability density function: $f(x) \ge 0$ for all real $x$ and $\int_{-\infty}^{\infty}f(x)dx=1$
  • define the probability as the area under the graph of the probability density function using the notation $P(X \le r)=\int_a^r f(x)dx$, where $f(x)$ is the probability density function defined on $[a,b]$
  • examine simple types of continuous random variables and use them in appropriate contexts
  • explore properties of a continuous random variable that is uniformly distributed
  • find the mode from a given probability density function
• Obtain and analyse a cumulative distribution function with respect to a given probability density function
  • understand the meaning of a cumulative distribution function with respect to a given probability density function
  • use a cumulative distribution function to calculate the median and other percentiles

MA-S3.3

• Identify the numerical and graphical properties of data that is normally distributed
• Calculate probabilities and quantiles associated with a given normal distribution using technology and otherwise, and use these to solve practical problems
  • identify contexts that are suitable for modelling by normal random variable, eg the height of a group of students
  • recognise features of the graph of the probability density function of the normal distribution with mean $\mu$ and standard deviation $\sigma$, and the use of the standard normal distribution
  • visually represent probabilities by shading areas under the normal curve, eg identifying the value above which the top 10% of data lies
• Understand and calculate the z-score (standardised score) corresponding to a particular value in a dataset
  • use the formula $z=\frac{x− \mu}{\sigma}$, where $\mu$ is the mean and $\sigma$ is the standard deviation
  • describe the z-score as the number of standard deviations a value lies above or below the mean
• Use z-scores to compare scores from different datasets, for example comparing students’ subject examination scores
• Use collected data to illustrate the empirical rules for normally distributed random variable
  • apply the empirical rule to a variety of problems
  • sketch the graphs of $f(x)=e^{−x^2}$ and the probability density function for the normal distribution $f(x)=\frac{1}{\sigma \sqrt{2 \pi}}e^{−\frac{(x−μ)^2}{2 \sigma ^2}}$ using technology
  • verify, using the Trapezoidal rule, the results concerning the areas under the normal curve
• Use z-scores to identify probabilities of events less or more extreme than a given event
  • use statistical tables to determine probabilities
  • use technology to determine probabilities 
• use z-scores to make judgements related to outcomes of a given event or sets of data