Estimating Population Proportions

5 min

Narrative

The purpose of this Warm-up is for students to compute the fraction of individuals whose responses fall in a specified category. This activity gives students time to think about how to compute these fractions from categorical data.

For the last two questions, students may debate whether to include the 10-minute times or not. According to the wording of the question asked, it does ask for more-than-10-minute times, so maybe exactly 10 minutes should not count (because 10 is not greater than 10). On the other hand, all of the values are listed as whole numbers, so a student who takes 10 minutes and 1 second to get to school may have rounded down to 10, but should have been counted. Noticing the large difference in answers for the third question, it may be worth clarifying the data in this instance, even for an estimate.

Monitor for students who include the 10-minute times for the last two questions as well as those who do not.

Launch

Give students 2 minutes of quiet work time, and follow with a whole-class discussion.

Student Task

A teacher asks all the students in one class how many minutes it takes them to get to school. Here is a list of their responses:

  • 20
  • 10
  • 15
  • 8
  • 5
  • 15
  • 10
  • 5
  • 20
  • 5
  • 15
  • 10
  • 3
  • 10
  • 18
  • 5
  • 25
  • 5
  • 5
  • 12
  • 10
  • 30
  • 5
  • 10
  1. What fraction of the students in this class say that:

    1. it takes them 5 minutes to get to school?
    2. it takes them more than 10 minutes to get to school?
  2. The whole school has 720 students. Use this data to estimate how many of them would say that it takes them more than 10 minutes to get to school.

    Be prepared to explain your reasoning.

Sample Response

    1. 724\frac{7}{24}
    2. 924\frac{9}{24} (or equivalent) or 1524\frac{15}{24} (or equivalent) if 10 minutes is included
  1. About 270 students (or 450 students if 10 minutes is included) since 924\frac{9}{24} of 720 is 270 (or 1524\frac{15}{24} of 720 is 450)

Synthesis

Select students to share their methods for computing the solutions. Include previously identified students who did or did not include the 10-minute values in their calculations.

If it does not arise during the discussion, explain that answering the last question with the data at hand is only accurate if the sample data is representative of the school. For example, it is possible that the class happens to contain only students who get a ride to school, but much of the school rides the bus.

Standards
Building Toward
  • 7.SP.1·Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
  • 7.SP.2·Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. <em>For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.</em>
  • 7.SP.A.1·Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
  • 7.SP.A.2·Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. <span>For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.</span>

15 min

15 min

10 min