Lesson 5 Would You Like to Try a Sample? Develop Understanding

Learning Focus

Understand methods for taking samples from a population.

What is a random sample?

Without asking everyone the questions, how can you be sure that a survey represents the opinions of an entire population?

Open Up the Math: Launch, Explore, Discuss

In the previous lesson, you saw a number of statistics for things like the average weight of a house cat. You know it would be impossible to measure all the house cats to find their average weights, but scientists still claim to know it. You’ve probably heard it many times before: “Survey results show that $54 %$ of Americans believe that…” You’re sure that you didn’t participate in the survey and neither did anyone you know, and yet, the researchers claim that the survey represents the beliefs of all Americans.

How can this be possible? In the next few tasks, we’ll explore how statistics allow us to draw conclusions about an entire group without actually working with the entire group. Sometimes the results make sense and other times, you might think that they just can’t be right. We will learn how to make judgments about statistical studies, based on the methods that have been used.

First, we need to get our terms straight. When we talk about the entire group that we are interested in, that is called the population. When some members of the group are selected to represent the entire group, that is called a sample. The thing we are interested in knowing about the population is the parameter of interest.

For each of the scenarios below, identify the population, the sample, and the population parameter of interest.

1.

A grocery store wants to know the average number of items that shoppers purchase in each visit to the store. They decide to count the items in the cart of every twentieth person through the check stand.

Population:

Sample:

Parameter of interest:

2.

A team of biologists wants to know the average weight of fish in a lake. They decide to drop a net and measure all the fish caught in three different locations in the lake.

Population:

Sample:

Parameter of interest:

3.

There are lots of different ways that a sample can be chosen from a population. Your teacher will be giving you a set of scenarios that show different ways to take a sample. Sort the scenarios into categories and be prepared to describe each category.

a.

You are in charge of school activities. You want to know what activities students would prefer to participate in during the school year. You decide to put the name of each student in the school into a big bowl. You draw $100$ names and ask those students to respond to a survey about the activities they prefer.

b.

You are in charge of school activities. You want to know what activities students would prefer to participate in during the school year. You assign each student in the school a number. You randomly select a starting number among the first $10$ numbers and then select every tenth student in the list from that point forward.

c.

You are in charge of school activities. You want to know what activities students would prefer to participate in during the school year. You use the rolls from each homeroom class. You go through each homeroom class, drawing $2$ names from each class. You ask those students to respond to a survey about the activities they prefer.

d.

You are in charge of school activities. You want to know what activities students would prefer to participate in during the school year. You get the list of all the homeroom classes and randomly select $5$ classes. You go to each of the classes selected and survey all the students in that class.

e.

You are in charge of school activities. You want to know what activities students would prefer to participate in during the school year. You stand in the cafeteria during your lunch break and ask students if they would be willing to participate in your survey as they walk by.

f.

You are in charge of school activities. You want to know what activities students would prefer to participate in during the school year. You make a lot of copies of the survey about the activities that students prefer and you put them on a table outside the cafeteria. Students can choose to take the survey and drop their responses into a big box on the table.

g.

You are interested in finding out the percent of residents in the city that have experienced a robbery in the past year. Using the city property records, you assign each residence a number. You use a random number generator to give you a list of numbers. You look up the police reports for each residence selected.

h.

You want to know the average number of hours that high school seniors spend playing video games in your state. You randomly select $20$ high schools in the state and then ask all the seniors at each of the $20$ high schools about their video game habits.

i.

An auto analyst is conducting a satisfaction survey, sampling from a list of $10,000$ new car buyers. The list includes $2,500$ Ford buyers, $2,500$ GM buyers, $2,500$ Honda buyers, and $2,500$ Toyota buyers. The analyst selects a sample of $400$ car buyers by randomly sampling $100$ buyers of each brand.

j.

A shopping mall management company would like to know the average amount that shoppers in the mall spend during their visit. They post two survey takers near one of the exits who ask shoppers to tell them what they spent as they leave the mall.

k.

A restaurant owner wants to find out the average number of dishes ordered at each table served on Friday evenings, their busiest time. She decides to collect and analyze every fifth receipt of the night, starting at 6:00 p.m.

l.

m.

n.

o.

4.

What might be some of the advantages and disadvantages of each type?

5.

A person you know works for the city in a small theater that shows local dramatic productions. She wants to know the average age of the people who buy tickets to see shows so that she can better select which plays to stage. She plans to ask the ages of the first twenty people who buy tickets for the show. What would be your advice to your friend?

6.

Describe a process for selecting a representative sample of the theater patrons.

Ready for More?

Besides the sampling method used, what other factors affect the validity of a survey?

Takeaways

Random Sampling Methods	Nonrandom Sampling Methods

Vocabulary

Lesson Summary

In this lesson, we learned about methods for obtaining samples from a population so that a parameter of interest can be studied. We learned that random sampling methods are preferable to other methods because they are more likely to represent the entire population.

Retrieval

Decide whether the variables simply explain each other (association) or if you think one variable would cause the other to change (causation).

1.

As the amount of ice cream sold per person increases, the amount of crime increases.

Does eating ice cream cause people to commit crimes? (both increase due to temperature)
Causation/ association?

2.

We didn’t have any eggs so we made the cake without them. It didn’t turn out.

Did leaving the two eggs out of the mix cause the cake to not turn out?
Causation/ association?

3.

Given: $f (t) = 12 \cos (\frac{π}{3} (t - 6)) - 5$ . Identify the amplitude, period, horizontal shift, and the vertical shift.

Amplitude:

Period:

Horizontal shift:

Vertical shift: