Lesson 2-I ~ Random Sampling 41

RANDOM SAMPLING

LESSON 2-I

Sometimes people want to know something about a large group of people and they cannot ask each person. For example, if Kira is running for seventh grade class president, she may want to know her chances of winning the election. Suppose her school has 800 seventh graders. Asking each person whether they will vote for Kira will take too much time. Instead, Kira needs to find a way to ask a smaller number of seventh graders and then use that information to predict whether or not she has a chance to win. But who should Kira survey in order to get an accurate idea of who will win the election when all the seventh graders vote?

Kira surveyed each of the following groups to find the percent of each group that would vote for her in the election.

40 seventh grade girls80% vote for Kira

40 seventh grade boys35% vote for Kira

40 students in Kira’s Math Class

75% vote for Kira

40 random seventh graders65% vote for Kira

Which group of students gives Kira the most accurate prediction?

Every time Kira asked 40 students who they would vote for, she was asking a sample of seventh graders instead of all the seventh graders. A sample is a smaller group that is used to make conclusions about the entire population. The population is the entire group. In the example above, the population is the group of 800 seventh graders that attend Kira’s school. The samples are each a group of 40 students that Kira surveyed.

42 Lesson 2-I ~ Random Sampling

The most accurate information that Kira will want to use in making her decision about running for class president will come from the sample that most closely resembles the entire seventh grade class. This is the representative sample. Kira needs to choose the best representative sample for an accurate prediction.

40 seventh grade girls80% vote for Kira

40 seventh grade boys35% vote for Kira

40 students in Kira’s Math Class

75% vote for Kira

40 random seventh graders65% vote for Kira

When Kira pulled 40 names from a hat, she selected a random sample of students. A random sample is created when every member of the population is equally likely to be chosen as part of the sample. Since the names of all 800 seventh graders were in the hat, each seventh grader had an equal chance of having his or her name drawn from the hat.

It is important to have a large enough sample that will be representative of the population. If Kira had only asked three people, she would not be able to make an accurate prediction. The more people she asks, the more accurate her prediction will be.

Sometimes samples are biased. A biased sample is created when people in a sample do not accurately represent the entire population. When Kira chose to ask only girls she created a biased sample. Both boys and girls will vote for class president so both need to be included in the sample. Even though it is possible for a random sample to have bias, it is not likely.

Margo wanted to know if people in her city would vote for a dog park. She could not ask everyone in the city so she decided she would call 50 random people from the list of people who take their animals to the vet.

Explain whether or not the sample of people surveyed will most likely give an accurate prediction of how people will vote in the election.

This sample will most likely NOT give an accurate prediction. People who go to the vet own animals and may be more likely to support a vote for a dog park. The sample needs to include a better representation from the entire city, both people who own animals and people who do not. It is a biased sample.

EXAMPLE 1

solution

Lesson 2-I ~ Random Sampling 43

A company is making a new video game. There is a possibility some of the games have a defect and do not work. The owner of the company wants to make sure the games work before he sells them. He has made 10,000 games and packed them into 100 boxes that each holds 100 games. He is trying to decide if he should test 100 games from a single box or if he should test 100 games by testing one game in each of the 100 different boxes.

Which sample do you think would give a more accurate prediction about the games and the possible defect?

Choosing a single game from each of 100 different boxes would most likely give a more accurate prediction of whether or not the games have a defect. Games from the same box are usually manufactured and packed at the same time. Games from different boxes were most likely made at different times and will better represent the entire population of games.

Step 1: Write down your favorite color of the four listed below on a piece of paper. This will be used to create samples and make a prediction about the most popular color in the class. RED BLACK GREEN BLUE

Step 2: Copy the chart to use for each sample in Steps 3-6.

Sample Red Black Green Blue Prediction for most popular color in the whole class

All boysAll girls

5 students15 students

Step 3: Ask the boys to stand and share their colors chosen. Fill in the table. Make a prediction.

Step 4: Ask the girls to stand and share their colors chosen. Fill in the table. Make a prediction.

Step 5: Pull 5 names from a hat and ask the students to stand and share their colors chosen. Fill in the table. Make a prediction.

Step 6: Pull 15 names from a hat and ask the students to stand and share their colors chosen. Fill in the table. Make a prediction.

Step 7: Count the number of colors chosen by the students in the entire class. Which color did the majority of students choose? Which sample predicted that color? Explain why a sample may or may not have predicted the color actually chosen by most of the students.

EXPLORE! FAVORITE COLOR

EXAMPLE 2

solution

44 Lesson 2-I ~ Random Sampling

EXERCISES

1. What is the difference between a sample and a population?

2. Will a random sample always accurately predict the outcome for the entire population? Why or why not?

3. How can you guarantee a prediction from a survey is accurate?

Identify each type of sample as representative or biased. Explain your reasoning in a sentence.

4. A survey about a “favorite cereal” is given to every fifth person who enters a supermarket.

5. Paul asks all his friends to identify their favorite item on the school lunch menu.

6. A movie studio surveys all the adults in an audience leaving a children’s movie to “rate the film”.

7. Jill randomly picks 100 people from a phone directory and asks them about their “favorite restaurant”.

Consider the given information and identify the best sample. Explain your reasoning.

8. David wants to know which football team is the most popular in the country. Which sample will give him the best chance to make an accurate prediction? A. A survey of 80 men walking into the stadium before the game. B. A survey of 80 men and women leaving the stadium after the game. C. A survey of 80 people randomly selected from a sporting goods company’s mailing list. D. A survey of 80 people randomly selected from a national phone directory.

9. Olga has just moved to a new town and she needs to find a dentist. Olga wants to choose the best dentist in town. What is the best way for Olga to determine her dentist? A. Post a survey on a website for a week. B. Distribute a survey to patients leaving their dentist appointments at one dentist location. C. Question every fourth person entering the local supermarket about their personal experience with their own dentist.

D. Ask a few of her new neighbors who they think is the best dentist in town.

Each sample used in a survey below is biased. Explain how to modify the sample to eliminate the bias.

10. Joe wants to know what percent of people are likely to go see the new comedy film that is showing at the local theater. He asks everyone that he finds renting comedies at the movie rental store if they are planning to go see the new movie.

Lesson 2-I ~ Random Sampling 45

11. David wants to determine which of the four pizza restaurants in his town is the most popular. He stands outside each restaurant and counts people coming and going for one hour. Since he has to move to four different

restaurants, David visits one restaurant at 1:00 pm, another restaurant at 3:00 pm, the third restaurant at 5:00 pm and the last restaurant at 7:00 pm. 12. Jose wants his dad to raise his weekly allowance. To determine how much he should ask for, Jose surveys all his friends to learn what their weekly allowance is.

Is each prediction as accurate as possible based on the sample surveyed? Explain why or why not.

13. A farmer wants to know if his trees have been infected with a virus. He tests two trees out of one thousand and neither of his trees is sick. He decides all of his trees must be fine.

14. Ms. Smie wants to know if her students did their homework. She asks all of the students in her homework club if they finished. Nearly all of the students in the homework club finished, so she assumes most of her students will have finished their homework.

15. Tom wants to know which candidate people in his town will choose for mayor. He surveys 50 people who live on his street. They all choose Karen. Tom predicts Karen will win the election.

16. Storm wants to know how many students at his school have cell phones. There are 200 students in his school. He asks every fourth student who enters the building until he reaches 50 students. Based on their answers, he determines about 70% of the students at his school have cell phones.

PIZZA