Statistics and Parameters

• Statistic: a number that describes some characteristic of the sample

  • Relevant symbols

    • Mean: x̄

    • Proportion: p̂

    • Standard deviation: S

• Parameter: a number that describes some characteristic of the population

  • Relevant symbols

    • Mean: μ

    • Proportion: p

    • Standard deviation: σ

• Samples are taken to try to estimate the population (μ or p)

• Ultimate goal: Estimate parameters based on statistics

Distribution, Variability, and Bias

• Sampling distribution: the distribution of values taken by the statistic in all possible samples of the same size from the same population

  • Eg. sample mean, proportion

• Sampling variability: how much results vary between samples

  • Every time a sample is taken, the results will vary

  • Measured using the spread of the random sample

  • Based primarily on the size of the random sample

• As a general rule of thumb,

  • Larger sample: less variability

  • Smaller sample: more variability

• The spread does not depend on the size of the population, as long as it is at least 10x larger than the sample

• Biased statistic: a statistic which consistently overestimates or underestimates the parameter (mean or proportion)

• Unbiased statistic: a statistic in which the distribution of samples is centered around the true population’s parameter

Bias vs. variance

Means and Proportions

• Proportion problems generally involve categorical variables

• Binomial distributions will become approximately normal distributions if np≥10 and n(1-p)≥10

  • n = sample size

  • p = population proportion

  • p̂ = sample proportion

  • Normal distribution means the use of z-scores as a standard measure

• Statistics are unbiased if they are equal to the true parameters, so p̂ is an unbiased estimator of p

Verifying Conditions

• If an SRS is taken of size n from a large population with proportion p,

  • Some conditions must be stated and checked

    • Is the population more than 10x larger than the sample size?

      • The 10% condition verifies that the standard deviation formula may be used

    • Is np ≥ 10 and is n(1-p) ≥ 10?

      • This verifies that a normal approximation may be used

• If these conditions are met, the mean of the sample proportions will equal the true population proportion