STA130: Lecture_6

Estimation and random selection §

• Over-dependence on p-values may cause invalid results in the scientific process

• The choice of $\alpha$ is arbitrary, it’s better to either comment on the strength of the evidence against $H_0$ by reporting the p-value, or at least make sure to choose a $\alpha$ before calculating the p-value

• 2 main branches of statistical inference

  1. Testing - hypothesis test (evaluate evidence against a particular value for parameter)

  2. Estimation - confidence interval (estimate of a parameter, gives range of plausible values of parameter)

  • Both based on statistics (estimates of parameters from sample) and sampling distributions (estimates of them of statistics)

• Goals of estimation:

  • Want to find something about a population, resultant value is the true value of the parameter
  • Can’t get to everyone, some take a random sample from the population to estimate
  • An estimate is just an estimate, to try to count for the fact that it is probably wrong, but hopefully close to the true value.

Estimates, parameters and many samples §

• The function sample_n() from dplry can be used to draw samples of any size form a data frame (default is to sample without replacement)

  sample25 <- <dataset> $>$ sample_n(size=25, replace = FALSE)

  • This function samples rows (observations) from a data frame, with or without replacement. While the sample() function samples elements from a vector, with or without replacement

• As the sample size increase:

  • The mean of the sampling distribution - stay roughly the same (approx. the parameter)
  • The shape of the sampling distribution - less skewed
  • The spread of the sampling distribution - less spread out

The Bootstrap Method §

• The sample must be representative of the full population

  

• Start with a sample of size n from the population (assume it is representative)

  1. Draw many bootstrap samples of size n (with replacement) from the original sample
  2. For each bootstrap sample, calculate the statistic

• The distribution of the values of the statistics for all bootstrap samples is the bootstrap sampling distribution (this is another estimate of the sampling distribution of the statistic)

• Note that bootstrapping does not create new data. It simply is a tool to allow us to explore the variability of estimates from our original sample.

	Bootstrapping	Hypothesis Test
Initial data production	Random sample from a population of interest	- Random sample from a population of interest AND assignment of units to treatment groups
re-sample	Random re-sample (with replacement) from the original sample	Random re-assignment of labels
Goal	Create a range of plausible values for the population parameter	Conclude if we can rule out the ‘chance-was-acting-alonee’ explanation because it’s implausible