## 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.

**Problem H1
**Find the average (mean) value for all 100 people. Computer software may make this process easier.

**Problem H2
a. **Use the random process you developed in Part B to generate a random sample of 10 people from the population. Sample without replacement.

**Calculate your sample mean and compare it to your population mean.**

b.

b.

**Problem H3**

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.

**Problem H4
**Determine the Five-Number Summary for the set of sample means from your random samples of size 10.

**Problem H5**

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.

**Problem H6**

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**

Problem H7

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.

**Suggested Readings:
**

**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.

**Download PDF File:**

Counting Penguins

Continued

**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