9B Preview

Preparing for the next class

In the next in-class activity, you will need to be able to identify whether a summary measure is a parameter or a statistic, determine the appropriate type of plot for a given dataset, simulate sample proportions of random samples of a given size from a population with known p, describe and interpret features of a sampling distribution of sample proportions, and calculate the standard deviation of a sample proportion.

The 2020 November presidential election in the United States had one of the highest voter turnout rates in recent history: 66.7% of the voting-eligible population voted for a candidate for president.[1]

Question 1

Is the value 66.7% a parameter or a statistic?

  1. Parameter, since it summarizes an entire population
  2. Parameter, since it summarizes a sample from the population
  3. Statistic, since it summarizes an entire population
  4. Statistic, since it summarizes a sample from the population

Question 2

True or false: If you take a simple random sample of 1,000 individuals from the U.S. voting-eligible population and ask each individual whether they voted in the 2020 November presidential election, 667 of them will answer yes.

Question 3

Suppose you plan to take a simple random sample of 10 individuals from the U.S. voting-eligible population and ask each individual whether they voted in the 2020 November presidential election.

  1. What would be the appropriate plot to display these data?
    1. Scatterplot
    2. Boxplot
    3. Bar graph
    4. Normal curve
  2. Do you think it is very likely for your sample proportion to be 0.6? Explain.
  3. Do you think it is very likely for your sample to have [latex]\hat{p} = 0.3[/latex]? Explain.

Go to the DCMP Sampling Distribution of the Sample Proportion tool at https://dcmathpathways.shinyapps.io/SampDist_Prop/. You will use this tool to simulate taking random samples of 10 individuals from the U.S. voting-eligible population:

  • Set the Population Proportion to [latex]p= 0.67[/latex].
  • Set the Sample Size to [latex]n = 10[/latex].
  • Simulate taking 1,000 random samples of size 10 by selecting “1,000” and clicking “Draw Sample(s).”

The “Sampling Distribution of Sample Proportion” graph at the bottom of the tool displays the distribution of the 1,000 sample proportions from your random samples.

Question 4

Use the “Sampling Distribution of Sample Proportion” graph to approximate the proportion of the simulated samples with [latex]\hat{p}= 0.6[/latex].

Question 5

Use the bar above [latex]\hat{p}= 0.5[/latex] on the “Sampling Distribution of Sample Proportion” graph to complete the following sentence:

For a sample of 10 individuals from the U.S. voting-eligible population, the approximate probability that exactly ______ individuals in the sample voted in the 2020 November presidential election is ______ .

Question 6

The standard deviation of simulated sample proportions should be close to 0.15. Which of the following is a correct interpretation of this value?

  1. We would expect a typical sample proportion of individuals who voted in the 2020 November presidential election to be around 0.15.
  2. We would expect a typical sample proportion of individuals who voted in the 2020 November presidential election in a random sample of size 10 to be about 0.15 away from 0.67.
  3. All sample proportions of individuals who voted in the 2020 November presidential election in random samples of size 10 will be 0.15 away from 0.67.
  4. About 50% of sample proportions of individuals who voted in the 2020 November presidential election in random samples of size 10 will be between 0.52 and 0.82.

Rather than using simulation, we can use mathematical theory to derive expressions for the mean and standard deviation of the sampling distribution of the sample proportion.

Keep these properties in mind as you answer the remaining questions in this preview assignment.

Sampling Distribution of the Sample Proportion

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

  • 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]. Type your textbox content here.

Question 7

Use a formula to calculate the standard deviation of sample proportions when taking random samples of size 10 from a population where [latex]p= 0.67[/latex].

 

  1. United States Elections Project. (2020, December 7). 2020 November general election turnout rates. http://www.electproject.org/2020g ↵