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Name__________________________________ Class__________________________ Date_______________________ 10-1

Data-Gathering TechniquesCOMMON CORE

CC.9-12.S.IC.1*Essential question: What are the different methods for gathering data about a population?

1 ENGAGE Understanding Data-Gathering Techniques

In the branch of mathematics known as statistics, you work with data. Data can be numerical, such as heights or salaries, or categorical, such as eye color or political

affliliation. You collect data about a population by surveying or studying some or all of the individuals in the population.

When all the individuals in a population are surveyed or studied, the data-gathering technique is called a census. A parameter is a number that summarizes a

characteristic of the population.

When only some of the individuals in a population are surveyed or studied, the data-gathering technique is called sampling. A statistic is a number that summarizes a

characteristic of a sample. Statistics can be used to estimate parameters. Samples that result in accurate estimates are said to be representative of the population.

There are a variety of sampling methods, characterized by how the individuals in the sample are chosen. The table below lists a few.

Sampling Method Description

Random Each individual in the population has an equal chance of being selected.

Self-selected Individuals volunteer to be part of the sample.

Convenience Individuals are selected based on how accessible they are.

SystematicMembers of the sample are chosen according to a rule, such as every nth individual in the population.

Stratified

The individuals are organized into groups, and individuals from each

group are selected (typically through a random sample within each

group).

Cluster

The individuals are organized into groups, and all of the individuals in just some of the groups are selected (typically through a random sample

of the groups).

REFLECT

1a. Give an example of numerical data and an example of categorical data other than the examples listed in the first paragraph.

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1b. Asking your friends is an example of what type of sampling method? Explain.

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1c. Which sampling method do you think is most likely to result in a representative sample? Why?

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1d. Which sampling method do you think would be least likely to result in a representative sample? Why?

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1e. Explain why a researcher might use a sampling method rather than a census to gather information about a population.

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2 EXPLORE Finding Statistics Using Various Sampling Methods

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The salaries (in thousands of dollars) of all 30 employees at a small company are listed in the table.

Salaries at a Small Company

21 24 26 28 30 32 33 35 37 41

44 46 47 49 50 51 52 54 55 57

58 62 62 64 64 65 70 71 73 80

Use the table to generate a sample of 6 individuals using each sampling method, and then use the sample to predict the mean of the population.

A Suppose individuals whose salaries are 51, 57, 58, 65, 70, and 73 volunteer to be in the sample. Compute the self-selected sample’s mean, rounding to the nearest

whole number.

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B Take a convenience sample by choosing the 6 numbers in the first two columns of the table. Record the salaries, and then compute the sample’s mean, rounding to

thenearest whole number.

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C Take a systematic sample by choosing every fifth number in the list, reading from left to right in each row. Record the salaries, and then compute the sample’s

mean, rounding to the nearest whole number.

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D Take a random sample. Begin by labeling the data in the table with the

identifiers 1–10 for the first row, 11–20 for the second row, and 21–30

for the third row. Then use a graphing calculator’s random integer

generator to generate 6 identifiers between 1 and 30, as shown. (If any

identifiers are repeated, simply generate replacements for them until

you have 6 unique identifiers.) Record the corresponding salaries, and

then compute the sample’s mean, rounding to the nearest whole

number.

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REFLECT

2a. Compute the mean of the population. Then list the four samples from best to worst in terms of how well each sample mean estimates the population mean.

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2b. How do the best and worst sampling methods from your list compare with your answers to Reflect Questions 1c and 1d?

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Some statistics, such as the mean, apply only to numerical data. For categorical data, an appropriate statistic is a proportion, which is the relative frequency of a

category.

3 EXAMPLE Making Predictions from a Sample

A community health center surveyed a small

random sample of adults in the community

about their exercise habits. The survey asked

whether the person engages in regular cardio

exercise (running, walking, swimming, or other)and, if so, what the duration and frequency of exercise are.

Of the 25 people surveyed, 10 said that they do

engage in regular cardio exercise. The table lists

Type of

exercise

Duration(minutes

spent exercising)

Frequency(times

per week)

Running 30 4

Walking 20 5

Running 40 3

Running 60 6

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the data for those 10 people.

Calculate statistics from the sample, and use the

statistics to make predictions about the exercise

habits of the approximately 5000 adults living in

the community.

Swimming 40 4

Other 90 2

Running 30 3

Walking 20 5

Running 30 4

Other 120 1

A A Calculate the following two proportions from the sample data.

Proportion of adults who get regular cardio exercise = = %

Proportion of runners among those who get regular cardio exercise = = %

B Use the proportions from part A and the verbal model below to predict the number of runners among all adults living in the community.

Number of

runners in the

community=

Number of

adults in the

community×

Proportion of

adults who get

cardio exercise×

Proportion of runners

among those who get

regular cardio exercise

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C Calculate the following two means from the sample data.

Mean duration of exercise for those who get regular cardio exercise = ____________________________

Mean frequency of exercise for those who get regular cardio exercise = ____________________________

D Use the means from part C to predict, for those who get regular cardio exercise, the number of hours spent exercising each week.

Show your calculations and include units.

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REFLECT

3a. One of the categorical variables in the survey was regular cardio exercise. That variable had only two possible values: yes or no. What was the

other categorical variable, and what were its possible values?

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3b. What were the two numerical variables in the survey, and what were their possible values?

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3c. How much confidence do you have in the predictions made from the results of the survey? Explain your reasoning.

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PRACTICE

A student council wants to know whether students would like the council to sponsor a mid-winter dance or a mid-winter carnival this year. Classify each

sampling method.

1. Survey every tenth student on the school’s roster. ___________

2. Survey all freshmen and all juniors. ___________

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3. Survey 20 freshmen, 20 sophomores, 20 juniors, and 20 seniors. ___________

4. Survey those who ask the council president for a questionnaire. ___________

5. Survey those who happen to be in the cafeteria at noon. ___________

Use the following information for Exercises 6–9.

The officers of a neighborhood association want to know whether residents are interested in beautifying the neighborhood and, if so, how much money they are willingto contribute toward the costs involved. The officers are considering the three sampling methods below.

A. Call and survey every tenth resident on the association’s roster.

B. Randomly select and survey 10 residents from among those who come to the neighborhood block party.

C. Mail a survey to every resident with instructions to complete and mail the survey back.

6. Identify the population.

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7.Which sampling method is most likely to result in a representative sample of the population? Explain.

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8. Describe another sampling method that is likely to result in a representative sample of the population.

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9. Describe the categorical and numerical data that the officers of the neighborhood association want to gather through a survey.

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Use the following information for Exercises 10–14.

A community theater association plans

to produce three plays for the upcoming

season. The association surveys a

random sample of the approximately7000 households in the community to see

if an adult member of the household is

interested in attending plays and, if so, what

type of plays the person prefers (comedy,

drama, or musical), how many members

of the household (including the person

surveyed) might attend plays, and how

many of the three plays those household

members might attend.

Of the 50 adults surveyed, 12 indicated an

interest in attending plays. The table liststhe data for those 12 people.

Preferred

type of play

Number of

people

attending

Number of

plays attending

Comedy 2 1

Musical 3 2

Musical 1 2

Drama 2 3

Comedy 3 2

Comedy 2 3

Musical 4 1

Drama 2 3

Comedy 2 2

Musical 2 3

Comedy 5 1

Drama 1 2

10. Describe the categorical and numerical data gathered in the survey.

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11. Calculate the proportion of adults who indicated an interest in attending plays. Then calculate the proportion of those interested in attending plays who prefer

dramas.

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12. Approximately 15,000 adults live in the community. Predict the number of adults who prefer plays that are dramas. Show your calculations.

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13. For an adult with an interest in attending plays, calculate the mean number of household members who might attend plays. Then calculate the mean number of plays

that those household members might attend. Round each mean to the nearest tenth.

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14. The theater association plans to sells tickets to the plays for $40 each. Predict the amount of revenue from ticket sales. Show your calculations and include units.

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