Sampling Variability: Learn It 3

  • Check the conditions for normal approximation of a sampling distribution of a sample proportion.
  • Use the normal distribution to calculate probabilities and percentiles from a sampling distribution.
  • Find the sample size needed for a sampling distribution to have a desired standard deviation.

Calculating Probabilities for the Sampling Distribution of the Sample Proportion

Sampling Distribution of the Sample Proportion

When taking many random samples of size [latex]n[/latex] from a population distribution with proportion [latex]p[/latex]:

  • The mean of the distribution of sample proportions is [latex]p[/latex].
  • The standard deviation of the distribution of sample proportions is [latex]\sqrt{\frac{p(1-p)}{n}}[/latex].
  • The normal condition states that if [latex]np\ge10[/latex] and [latex]n(1-p)\ge10[/latex] then the sampling distribution is approximately normal by the Central Limit Theorem. 

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When the sample size is large enough, we can use [latex]\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}[/latex] in place of[latex]\sqrt{\frac{p(1-p)}{n}}[/latex].  This is called the standard error, which is the estimated standard deviation of sample proportions.