Sampling Distribution of a Sample Proportion: Fresh Take

  • Use technology to create a sampling distribution of a sample proportion given [latex]n[/latex] and [latex]p[/latex].
  • Calculate the mean and standard deviation for a sampling distribution of a sample proportion.
  • Recognize the difference between the standard deviation and the standard error of a sample proportion.

Every day, we often see articles in the newspaper reporting the result of a poll using proportions or percentages.

[1]The Cato Institute 2022 Student Debt Cancellation National Survey was designed and conducted by the Cato Institute in collaboration with YouGov. The title of the article is “76% of Americans Oppose Student Debt Cancellation if It Drives up the Price of College, 64% Oppose if It Raises Taxes“.

However, did you read the fine print? Or, in this case, the bottom of the article? The margin of error for the survey of [latex]2000[/latex] Americans [latex]18[/latex] years of age and older is [latex]+/- 2.39[/latex] percentage points at the [latex]95\%[/latex] level of confidence.

We always need to be aware of the headline of an article and not take it as a factual information. Most of the information we obtained are from a sample and therefore we can only make inference about the population.

Now, to calculate the margin of error, we need to first understand how to find the mean of the distribution and how to calculate interpret the standard error.

  • The estimated mean of the distribution of sample proportions is [latex]\hat{p}[/latex].
  • To distinguish it from the true standard deviation of sample proportions, we call the estimated standard deviation of sample proportions the standard error of [latex]\hat{p}[/latex]:

Standard Error: [latex]SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}[/latex]

Suppose that a market research firm is hired to estimate the percent of adults living in a large city who have cell phones. Five hundred randomly selected adult residents in this city are surveyed to determine whether they have cell phones. Of the [latex]500[/latex] people surveyed, [latex]421[/latex] responded yes – they own cell phones. Find the mean and standard deviation of the sampling distribution of this sample proportion.

  1. https://www.cato.org/blog/new-poll-76-americans-oppose-student-debt-cancellation-it-drives-price-college-64-oppose-it ↵