Learning Math: Data Analysis, Statistics, and Probability
Random Sampling and Estimation Homework
These homework problems will take you through the process of statistical estimation using numerical data. You will investigate the quality of the estimation procedure based on different sample sizes.
Gather numerical data for 100 different people. Gather data that has a significant amount of variation; for example, the age of 100 fourth-grade students would not be good numerical data for these purposes. Other than that, the data can be very simple, such as height in centimeters or age in years.
These 100 people will represent your overall population for the seven homework problems. Your goal is to investigate your sample mean as an estimate of your population mean and to explore the accuracy of your estimates.
Solutions are not provided for these homework problems, since answers will vary according to the data you have gathered.
Find the average (mean) value for all 100 people. Computer software may make this process easier.
a. Use the random process you developed in Part B to generate a random sample of 10 people from the population. Sample without replacement.
b. Calculate your sample mean and compare it to your population mean.
Repeat the process of generating a random sample of size 10 and calculating the sample mean at least nine more times. Computer software may make this process easier and allow you to take more samples.
Determine the Five-Number Summary for the set of sample means from your random samples of size 10.
You will now generate a new set of estimates, this time based on random samples of size five.
a. Do you expect these estimates to have more or less variation than the estimates from samples of size 10?
b. Generate several random samples of size five. Use the same random process and generate the same number of random samples (at least 10) that you did in Problem H3. Determine the mean for each sample.
c. Determine the Five-Number Summary for these estimates.
Compare the estimates from samples of size 10 with the estimates from samples of size five. Draw comparative box plots. Where is the population mean in relation to each box plot? Which sample size produces estimates with less variation? Is this what you predicted?
TAKE IT FURTHER
a. Generate the same number of random samples that you’ve been using, but this time of size 20.
b. Use computer software to compute the average and standard deviation of all the sample means of size five and all the sample means of size 20. Compare the results: Which set has the smaller standard deviation? How much smaller is it?
You may need a large number of samples to see a pattern here.
Perry, Mike and Kader, Gary (February, 1998). Counting Penguins. Mathematics Teacher, 91 (2), 110-116.
Reproduced with permission from Mathematics Teacher. Copyright © 1998 by the National Council of Teachers of Mathematics. All rights reserved.
Woolley, Thomas (Autumn, 1998). A Note on Illustrating the Central Limit Theorem. Teaching Statistics, 20 (3), 89-90.
This article first appeared in Teaching Statistics <http://science.ntu.ac.uk/rsscse/ts/> and is used with permission.
Download PDF File:
A Note on Illustrating the Central Limit Theorem
Session 1 Statistics As Problem Solving
Consider statistics as a problem-solving process and examine its four components: asking questions, collecting appropriate data, analyzing the data, and interpreting the results. This session investigates the nature of data and its potential sources of variation. Variables, bias, and random sampling are introduced.
Session 2 Data Organization and Representation
Explore different ways of representing, analyzing, and interpreting data, including line plots, frequency tables, cumulative and relative frequency tables, and bar graphs. Learn how to use intervals to describe variation in data. Learn how to determine and understand the median.
Session 3 Describing Distributions
Continue learning about organizing and grouping data in different graphs and tables. Learn how to analyze and interpret variation in data by using stem and leaf plots and histograms. Learn about relative and cumulative frequency.
Session 4 Min, Max and the Five-Number Summary
Investigate various approaches for summarizing variation in data, and learn how dividing data into groups can help provide other types of answers to statistical questions. Understand numerical and graphic representations of the minimum, the maximum, the median, and quartiles. Learn how to create a box plot.
Session 5 Variation About the Mean
Explore the concept of the mean and how variation in data can be described relative to the mean. Concepts include fair and unfair allocations, and how to measure variation about the mean.
Session 6 Designing Experiments
Examine how to collect and compare data from observational and experimental studies, and learn how to set up your own experimental studies.
Session 7 Bivariate Data and Analysis
Analyze bivariate data and understand the concepts of association and co-variation between two quantitative variables. Explore scatter plots, the least squares line, and modeling linear relationships.
Session 8 Probability
Investigate some basic concepts of probability and the relationship between statistics and probability. Learn about random events, games of chance, mathematical and experimental probability, tree diagrams, and the binomial probability model.
Session 9 Random Sampling and Estimation
Learn how to select a random sample and use it to estimate characteristics of an entire population. Learn how to describe variation in estimates, and the effect of sample size on an estimate's accuracy.
Session 10 Classroom Case Studies, Grades K-2
Explore how the concepts developed in this course can be applied through a case study of a K-2 teacher, Ellen Sabanosh, a former course participant who has adapted her new knowledge to her classroom.
Session 11 Classroom Case Studies, Grades 3-5
Explore how the concepts developed in this course can be applied through case studies of a grade 3-5 teacher, Suzanne L'Esperance and grade 6-8 teacher, Paul Snowden, both former course participants who have adapted their new knowledge to their classrooms.