Sampling Variability: Apply It 2

  • 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.

Let’s use the statistical tool below to simulate samples of different sizes from the American adult population, where the sample proportion who are obese is calculated for each sample.

Let’s conduct a study to decide if there is evidence that the rate of obesity among the clinic’s patients is less than [latex]42.4\%[/latex].You will need to check the Enter Numerical Values for [latex]n[/latex] and [latex]p[/latex] box to enter the value [latex]n[/latex] and [latex]p[/latex].Observe the center, spread, and shape of the sampling distribution as the sample size increases as instructed below.

  • Set the sample size to [latex]n=1[/latex]. Then draw [latex]1,000[/latex] random samples of size [latex]1[/latex] from the population.
  • Set the sample size to [latex]n=5[/latex] and click Reset. Draw [latex]1,000[/latex] random samples of size [latex]5[/latex] from the population.
  • Set the sample size to [latex]n=25[/latex] and click Reset. Draw [latex]1,000[/latex] random samples of size [latex]25[/latex] from the population.
  • Set the sample size to [latex]n=100[/latex] and click Reset. Draw [latex]1,000[/latex] random samples of size [latex]100[/latex] from the population.

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In the question above, you witnessed the Central Limit Theorem at work.

The Central Limit Theorem states that, as the sample size gets larger, the distribution of the sample proportion will become closer to a normal distribution.

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].
  • If [latex]np\geq 10[/latex] and [latex]n(1-p) \geq 10[/latex], then the Central Limit Theorem states that the distribution of the sample proportions follows an approximate normal distribution with mean p and standard deviation [latex]\sqrt{\frac{p(1-p)}{n}}[/latex]