Preview Activity: Sampling Distribution of the Proportion

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Handout Preview Activity: Sampling Distribution of the Proportion

Now go to the GeoGebra link below to investigate the sampling distribution of a proportion.

Link to GeoGebra: https://www.geogebra.org/m/srwskfpv

Try to develop rules based on your observations.

Exploration:
- Experiment 1: Varying Sample Size with \(p=0.5\text{.}\) Set the population proportion to \(p=0.5\text{.}\) Try several different sample sizes and run as many trials as is helpful to see the shape of the distribution. (I recommend somewhere from 100 to 300 trials.)
  
  \(p\) \(n\) Shape of Distribution Center (around what value?) Spread (wide/narrow)
  
  \(0.5\) \(10\)
  
  \(0.5\) \(30\)
  
  \(0.5\) \(50\)
  
  \(0.5\) \(100\)
  
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- Experiment 2: Varying Proportion with \(n\) Fixed. Now fix \(n=50\text{.}\) Try several different values of \(p\) and run as many trials as is helpful to see the shape of the distribution. (I recommend somewhere from 100 to 300 trials.)
  
  \(p\) \(n\) Shape of Distribution Center (around what value?) Spread (wide/narrow)
  
  \(0.1\) \(50\)
  
  \(0.3\) \(50\)
  
  \(0.5\) \(50\)
  
  \(0.7\) \(50\)
  
  \(0.9\) \(50\)
  
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- Experiment 3: Checking the Rule of Thumb. For each combination you tried above, check whether the conditions below hold:
  
  \begin{equation*} np\geq 5\text{ and } n(1-p)\geq 5 \end{equation*}
  
  \(p\) \(n\) \(np\) \(n(1-p)\) Did it look normal in the applet?
  
  0.5 10
  
  0.5 30
  
  0.5 50
  
  0.5 100
  
  0.1 50
  
  0.3 50
  
  0.5 50
  
  0.7 50
  
  0.9 50
  
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Analysis Questions:
1. For \(p=0.5\text{,}\) how does the shape of the sampling distribution change as \(n\) increases?
  
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2. How does the spread (variability) of the distribution change as \(n\) increases?
  
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3. For a fixed \(n=50\text{,}\) how does changing \(p\) affect the distribution?
  
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4. Compare your observations with the rule of thumb in Experiment 3 -- did the distribution look normal when those conditions were met?
  
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