Mathematics: Statistics 2

1. Probability density functions
   1. To find mean and variance
   2. Consideration of area
2. Understand concept

Annotations:

• x~N(\(\mu\),\(\sigma^2\))

1. Solve problems
   1. P(X>x)
   2. Relationships between variables

      Annotations:

      • \(x\) , \(\mu\) , \(\sigma\)
2. Use tables
   1. Or calculator functions
3. Models continuous random variables
4. Approximation to binomial
   1. Conditions
   2. Continuity Correction

1. Understand conditions of distribution
2. Mean and variance

   Annotations:

   • X~Po(\(\mu\)) both mean and variance = \(\mu\)
3. Use normal approximation when needed
   1. Continuity correction
   2. Conditions of aproximation
4. Approximation to binomial
   1. Conditions
5. Calculate probabilities
   1. Using tables
   2. Using formula

1. Understand differences between population and sample
   1. Importance of randomness in samples
2. Producing random samples
   1. eg. random numbers
3. Recognise sample mean is random variable

   Annotations:

   • Sample \(\mu\) = Population \(\mu\) Sample \(\sigma^2\) = \(\frac{Population \sigma^2}{n}\)
4. How distribution of population and sample are related
5. Central Limit Theorem
6. Calculate unbiased estimates of population mean and variance from a sample
7. Understand hypothesis tests
   1. One-tailed and two-tailed tests
   2. Null hypothesis
   3. Alternative hypothesis
   4. Significance level
   5. Rejection region/Critical region
   6. Acceptance region
   7. Test statistic
8. Formulate hypotheses and carry out tests
   1. Single observation from a binomial distribution
      1. Perhaps using normal approximation
   2. Normal distribution
   3. A large sample using CLT
   4. Single observation from a Poisson distribution
9. Type I and type II errors