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Sampling Distributions

Sampling Distributions for Sample Means

Take all possible samples of size nn from a population and calculate the sample mean x‾\overline{x} for each, all possible values of the sample mean are calculated

Parameters of Sampling Distributions for Sample Means

  • Mean = μ\mu
  • Standard deviation = σn\dfrac{\sigma}{\sqrt{n}}
  • Standardized zz-score = x‾−μσn\dfrac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}

Normality of Sampling Distributions for Sample Means

  • If the population is normally distributed, the sampling distribution for sample means is normally distributed

Central Limit Theorem

  • If a population is not normally distributed
  • A large enough random sample of size n≥30n \geq 30 is taken while sample values are independent
  • Then the sampling distribution for sample means is approximately normally distributed

Sampling Distributions for Differences in Sample Means

One-Sample Problem

When one random sample of size nn has been taken from one population

Two-Sample Problem

If one random sample of size n1n_1 is taken from one population, then a different random sample of size n2n_2 is taken from a different population that is independent to the first population

Parameters of Sampling Distribution for Differences in Sample Means

  • Mean = μ1−μ2\mu_1 - \mu_2
  • Standard deviation = σ12n1+σ22n2\sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}}
    • Sampling with replacement or each sample size is less than 10% of the population size
    • Otherwise, σ\sigma will be smaller
  • Standardized zz-score = (x1‾−x2‾)−(μ1−μ2)σ12n1+σ22n2\dfrac{(\overline{x_1} - \overline{x_2}) - (\mu_1 - \mu_2)}{\sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}}}

Normality of Sampling Distributions for Differences in Sample Means

  • If two independent populations are normally distributed, the sampling distribution for differences in sample means is also normally distributed

Sampling Distributions for Sample Proportions

  • Population proportion, pp, is the percentage of success
  • nn is the sample size
  • XX is the number of successes in a sample, following binomial distribution
  • Sample proportion p^=Xn\hat{p} = \dfrac{X}{n}

If

  • np≥10np \geq 10
  • n(1−p)≥10n(1-p) \geq 10

Then p^\hat{p}

  • Normally distributed
  • Mean = pp
  • Standard deviation = p(1−p)n\sqrt{\dfrac{p(1-p)}n}
  • zz-score = p^−p(1−p)n\dfrac{\hat{p} - p}{{\sqrt{\dfrac{(1-p)}{n}}}}

Sampling Distributions for Differences in Sample Proportions

If

  • n1p1≥10n_1p_1 \geq 10
  • n1(1−p1)≥10n_1(1-p_1) \geq 10
  • n2p2≥10n_2p_2 \geq 10
  • n2(1−p2)≥10n_2(1-p_2) \geq 10

Then p1^−p2^\hat{p_1} - \hat{p_2}

  • Normally distributed
  • Mean = p1−p−2p_1 - p-2
  • Standard deviation = p1(1−p1)n1+p2(1−p2)n2\sqrt{\dfrac{p_1(1-p_1)}{n_1} + \dfrac{p_2(1-p_2)}{n_2}}
  • Standardized zz-score = (p1^−p2^)−(p1−p2)p1(1−p1)n1+p2(1−p2)n2\dfrac{(\hat{p_1} - \hat{p_2}) - (p_1 - p_2)}{\sqrt{\dfrac{p_1(1-p_1)}{n_1} + \dfrac{p_2(1-p_2)}{n_2}}}

Biased & Unbiased Estimators

Estimator is used to estimate the population parameter

To know if an estimator is a good predictor

  • All possible estimates from all possible samples of size nn must be generated
  • Check to see if, on average, estimates are centered around the value of the population parameter
  • An estimator is said to be unbiased if the mean of its sampling distribution equals the population parameter being estimated

Unbiased Estimators

  • Sample mean x‾\overline{x}
  • Sample proportion p^\hat{p}
  • Sample standard deviation ss

Affects on Unbiased Estimators

  • nn does not affect x‾\overline{x}
  • Greater nn gives more accurate S=σnS = \dfrac{\sigma}{\sqrt{n}}