Chapter 14: Censuses, Surveys, Polls and Studies Enumeration

Transcription

1 Chapter 14: Censuses, Surveys, Polls and Studies 14.1 Enumeration

2 Bell Work Take out a blank piece of lined paper and a writing utensil. #1 Statistics are about. #2 Did the Saudi Arabian people largely underestimate or overestimate the percentage of overweight/obese people in their country? #3 What would you say is the main idea of the TED Talk we watched yesterday? #4 What was one statistic you personally underestimated or overestimated in the online quiz we saw yesterday? What might be a reason for this? Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e:

3 Population and N-Value Enumerated = Population the term used to describe the individuals being counted. The individuals can be people, animals, plants, objects, etc. A population is well-defined when we can determine if the individuals belong to the given population. N-value the number of individuals in the population Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e:

4 Finding the N-value Finding the exact N-value of a large and elusive population can be extremely difficult and sometimes impossible. A census is an official count of the entire population. When would a census be difficult/impossible? In many cases, a good estimate is all we really need, and such estimates are possible through sampling methods. Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e:

5 Surveys for Enumeration Survey a data collection strategy that uses a sample to draw inferences about a population Sample a subset of the population chosen to be the providers of information in a survey Sampling the act of selecting a sample Statistic a numerical estimate of some measureable characteristic of a population obtained from a sample (usually an educated guess) Parameter a true measurement of some characteristic of a population, typically an unknown quantity Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e:

6 Example 14.4 Page 419 Defective Lightbulbs N 100,000 In general k n N P Where n = number in sample, k = # of individuals in sample having desired characteristic, P = total pop, N = # of individuals in pop w/ desired characteristic Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e:

7 Example As part of a sixth-grade class project the teacher brings to class a large jar containing 200 gumballs of two different colors: red and green. Andy is asked to draw a sample of his own choosing and estimate the number of red gumballs in the jar. Andy draws a sample of 25 gumballs, of which 8 are red and 17 are green. Use Andy s sample to estimate the number of red gumballs in the jar. Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e:

8 THE CAPTURE-RECAPTURE METHOD Step 1. Capture (sample): Capture (choose) a sample of size n 1, tag (mark, identify) the animals (objects, people), and release them back into the general population. Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e:

9 THE CAPTURE-RECAPTURE METHOD Step 2. Recapture (resample): After a certain period of time, capture a new sample of size n 2 and take an exact head count of the tagged individuals (i.e., those that were also in the first sample). Call this number k. Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e:

10 THE CAPTURE-RECAPTURE METHOD Step 3. Estimate: The N-value of the population can be estimated using the following proportion k n 1 n 2 N Where k = # tagged individuals in second sample n 2 = # of individuals in second sample n 1 = # of individuals in first sample N = what we are estimating Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e:

11 Example: Small Fish in a Big Pond A large pond is stocked with catfish. As part of a research project we need to estimate the number of catfish in the pond. An actual head count is out of the question (short of draining the pond), so our best bet is the capture-recapture method. Step 1. For our first sample we capture a predetermined number n 1 of catfish, say n 1 = 200. The fish are tagged and released unharmed back in the pond. Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e:

12 Example: Small Fish in a Big Pond Step 2. After giving enough time for the released fish to mingle and disperse throughout the pond, we capture a second sample of n 2 catfish. While n 2 does not have to equal n 1, it is a good idea for the two samples to be of approximately the same order of magnitude. Let s say that n 2 = 150. Of the 150 catfish in the second sample, 21 have tags (were part of the original sample). Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e:

13 Example: Small Fish in a Big Pond Assuming the second sample is representative of the catfish population in the pond, the ratio of tagged fish in the second sample (21/150) is approximately the same as the ratio of tagged fish in the pond (200/N). This gives the approximate proportion k n 1 n 2 N N Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e:

14 14.1 Assignment (pg. 438) #2-8 Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: