Statistics minor

Sampling

Necessary if:

  • The population is too large or a census is too expensive
  • The sampling process is destructive

Features of a sample:

  • Unbiased
  • Representative of the population
  • Data should be relevant
  • Data should not be affected by the act of sampling

Advantage of random sampling:

  • Enables proper inference to be undertaken; the probability basis under which the sample was taken is known
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Discrete random variables

Expectation and variance:

  • Var (X) = E (X^2) - E (X)
  • E (a + bX) = a + bE (X)
  • Var (a + bX) = b^2 Var (X)
  • E (aX +/- bY) = aE (X) +/- bE (Y)
  • Var (aX +/- bY) = a^2Var (X) + b^2Var (Y)

Conditions of Binomial distribution (also Geometric):

  • Each trial results in one of two outcomes
  • The probability of success is constant
  • The trials are independent of each other

Conditions of Poisson distribution:

  • Events occur randomly at a constant average rate, independently of each other

Po (lamda) + Po (mu) = Po (lamda + mu), if lamda and mu are independent

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Bivariate Data

Conditions of Pearson's product moment correlation coefficient:

  • Data should be random on random
  • Data should be from a bivariate Normal distribution
  • If one of the distributions is skewed, bimodal etc. it is unlikely to be appropriate

Null hypothesis: There is no correlation between ... and ...

Spearman's rank null hypothesis: There is no association in the population

Spearman's vs Pearson's:

  • Spearman's not appropriate if scatter diagram doesn't indicate a monotonic relationship (proportion)
  • Ranking data loses information

Regression lines: residual = observed value - value from regression line

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Chi-squared tests

Putting information into categories loses information

Contingency tables:

  • Null hypothesis: no association between ... and .../variables are independent
  • Calculate degrees of freedom (minimum number of values to work out the rest)
  • Table of expected values 
  • Chi-squared values table
  • Sum these to find the Chi-squared value

Goodness-of-fit:

  • Null hypothesis: The given model fits the data
  • Calculate degrees of freedom
  • Table of expected values - use model
  • Chi-squared values
  • Sum to find the Chi-squared value
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