# 4.3: Random Selection

**At Grade**Created by: Bruce DeWItt

### Learning Objectives

- Obtain a random sample using a random digit table
- Describe the process followed to obtain an SRS
- Outline an appropriate sampling method

### Random Selection

We have discussed that it is important to choose samples randomly in order to reduce bias, but we haven't discussed how to actually carry out the process. There are many ways to make random selections. A common way to choose things at random is to use a 'big hat', or box, or bowl, etc. For example, suppose that a teacher wanted to randomly select 5 students every day, from a class of 34 students, to hand in their homework to be graded. Each day she has all of the students' names in a big fish bowl. She will mix the names up and select 5 names. These students will turn their homework papers in right then, and the other students will not need to. The five selected names will be put back in the fishbowl, so they may be selected again tomorrow. This is an example of an SRS of size 5 of her class. Every student has an equal probability (5/34 or 14.7% chance) of being required to turn in his or her homework on any given day and any combination of five students may be chosen. One student may end up turning in her assignment several days in a row, while another student may never need to turn hers in all year long. The idea of a 'big hat' is a good method for random selection when working with small populations, but it is not always practical.

Random selections can be made by flipping coins, rolling dice, or spinning a spinner. These days, most random selections can be done using technology such as a computer program or a **random number generator** on a graphing calculator, Another way that random selections are made in statistics is by using a random digit table. A **random digit table** is a long list of randomly generated digits from 0 to 9. The digits are listed in groups of five simply to make it easier to read and not lose your place. Imagine that someone has a ten-sided die with each digit from 0 to 9 marked on a side. They sit down, roll the die and write down the digit that appears, then they roll it again and write down the digit that appears, then they do this again and again. As you can imagine, this would take quite awhile, but would result in a long list of random digits. This is basically what a random digit table is. There is a random digit table in the appendix for you to use.

### How to Use a Table of Random Digits

There is a process to follow when using a random digit table to make your selection. You need to report your process with enough detail that if someone else were to follow your steps, they would end up with the exact same randomly selected numbers. The purpose of this is to prove, if needed, that your selection process was truly random so that no one can accuse you otherwise. The following example illustrates the steps you will need to follow (and explain) when using a random digit table to make your random selection. **The random digit table can be found in the appendix.**

#### Example 1

Five boxes, each containing 24 cartons of strawberries, are delivered in a shipment to a grocery store. The produce manager always selects a few cartons randomly to inspect. He knows better than to just look at some of the cartons on the top or only in one box, because sometimes the rotten ones are on the bottom. Today he wishes to select a total of 6 cartons to inspect. He has the boxes arranged in order and has a set way to count the cartons inside each box. Explain the process used to make the random selection using a random digit table.

#### Solution

**Step 1: Assign numbers to the list (must all be an equal number of digits long)**

Since he has 120 cartons total, he will assign the numbers 001 to 120 to represent the cartons in order.

**Step 2: Choose a starting line on the random digit table. If the problem states a line to start at, use that line. Otherwise, pick any line you want and record the line number. If you run out of digits, simply move to the next line down.**

He will use line 119 to make the selections.

**Step 3: Decide how many digits to look at each time. The number of digits in your largest number is required.**

He will need to look at 3-digit numbers every time.

**Step 4: Decide if any numbers will need to be ignored and whether or not repeats will be allowed.**

He will not want to inspect the same carton twice, so he will ignore any repeats. And, any numbers above 120 will not apply in this case, so he will ignore numbers 121-999 and 000.

**Step 5: State when to stop.**

He will stop once six numbers are selected. He will then find the cartons that the numbers represent and inspect those cartons.

**Step 6: Report the numbers that were selected. When given a specific list, go back and determine which specific individuals have been selected.**

Here is a part of the random digit table so that you can see how the selection was made. Note that dividers have been placed between each group of 3-digits for this example. When we reach the end of a line, we simply continue on the following line.

*So, the strawberries in cartons numbered 042, 119, 025, 052, 087, and 072 will be inspected. The entire deliverywill be accepted or rejected based on this random sample of 6 cartons.*

#### Example 2

Five of the employees at the Stellar Boutique are going to be selected to go to a training in Las Vegas for four days. Everyone wants to go of course, so the owner has decided to make the selection randomly. She has decided to send two managers and three sales representatives. The employees' names are listed in the table below.

a) What type of sampling method is this?

b) Explain the process she can follow to use a random digit table, starting at line #108, to select the employees who will get to go to the training. Select the managers first, then select the sales representatives.

#### Solution

a) This is a stratified random sample.

b)

For the managers:

Assign numbers to the list 1 to 8Use random digit table, starting at line #108Look at one digit at a timeIgnore 9, 0, and any repeatsStop when two have been selectedState the names

So, Rosie(#6) and Gigi(#4) will be the managers who get to go to Las Vegas.

**Solution continued:**

For the sales representatives:

Assign numbers to the list 01-21Use random digit table, starting on the next line, #109Look at two digits at a timeIgnore 22-99, 00, and any repeatsStop when three have been selectedState the names

So, Ray Anne(#15), Sandy(#16), and Tawanda(#19) will be the sales representatives who get to go to Las Vegas.

### Problem Set 4.3

#### Section 4.3 Exercises

*Use the table of random digits in the appendix for the following problems.*

1. The manager at Big-N-Nummy-Burger wishes to know his employees' opinions regarding the work environment. He has 56 employees and plans to select 12 employees at random to complete a survey.

a) Explain the process he can follow to use a random digit table, starting at line 108, to select an SRS of size 12.

b) Which employees numbers were selected?

2. Use a random digit table to select an SRS of five of the fifty U.S. States. Explain your process thoroughly and report the five states that you chose. Repeat this a second time, but begin on a different line on the random digit table. Compare your lists to another classmate's lists. Did you end up with any of the same states in your samples?

3. Washington High School has had some recent problems with students using steroids. The district decides that it will randomly test student athletes for steroids and other drugs. The boy's hockey team is to be tested. There are 13 players on the varsity team and 21 players on the junior varsity team. Use a table of random digits starting at line 122, to choose a stratified random sample of 3 varsity players and 5 junior varsity players to be tested. Remember to clearly describe your process.

#### Review Exercises

4) Sketch a Venn Diagram that shows two events that are mutually exclusive.

5) Suppose that a survey was conducted at SRHS and it found that 86% of students have their own cell phones, and that 64% of students have their own IPod (or other similar personal music device). Furthermore, 9% of the students at SRHS say that they have neither one of these.

a) Define your variables and construct a Venn Diagram that fits this scenario.

b) What is the probability that a randomly selected student has both a cell phone and an IPod?

c) What is the probability that a randomly selected student has either a cell phone or an IPod?

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