Exercises: Use data from a random sample to draw inferences about a population - Common Core Math - Grade 7

A

Recall / Warm-Up

1.

In a survey, 26 out of 40 randomly selected students say they have a pet. What proportion of the sample has a pet?

A.

$0.65$

B.

$0.26$

C.

$0.40$

D.

$0.14$

2.

A teacher wants to estimate the favorite sport of all 7th graders in her school. Which sample is most likely to give a representative estimate?

A.

Ask the 12 students sitting near her at lunch

B.

Survey all students in her gym class

C.

Randomly select 30 names from the school roster

D.

Post a sign-up sheet for anyone who wants to respond

3.

A pollster randomly calls 200 registered voters and finds that 130 prefer Candidate A. What percent of the sample prefers Candidate A?

B

Fluency Practice

1.

A random sample of 40 students shows that 26 have pets. The school has 800 students. Estimate how many students in the school have pets.

2.

A factory's quality control team randomly tests 100 widgets and finds 4 are defective. The factory produces 10,000 widgets per day. About how many widgets per day are estimated to be defective?

3.

A student draws 5 random samples from the same population and gets these proportions: 0.50, 0.55, 0.60, 0.65, 0.60. Which statement best describes these results?

A.

The variation means at least one sample was collected incorrectly.

B.

The variation is normal — different random samples naturally produce different estimates.

C.

All 5 samples should have given exactly the same proportion.

D.

The sample size must have been too small to get consistent results.

4.

A researcher randomly surveys 80 city residents and finds 52 use public transit regularly. Which statement CORRECTLY reports this finding as a population inference?

A.

65% of all city residents use public transit regularly.

B.

Based on our sample, we estimate that about 65% of all city residents use public transit regularly.

C.

Exactly 65 out of every 100 city residents use public transit regularly.

D.

The survey proves that most city residents use public transit.

5.

A librarian randomly selects 50 books from a collection of 2,000. She finds 12 need repairs. Estimate how many books in the full collection need repairs.

C

Varied Practice

D

Word Problems

1.

A school is holding a student council election between two candidates. A pollster randomly calls 500 of the 12,000 registered student voters and finds that 310 prefer Candidate A.

1.

What proportion of the surveyed voters prefer Candidate A? Express your answer as a percent.

2.

Write an inference statement estimating how many of the 12,000 registered voters prefer Candidate A. Your statement must use appropriate hedging language to show this is an estimate.

2.

A reporter stood outside a grocery store and asked 15 shoppers whether they shop there weekly. Thirteen said yes. The reporter concluded that 87% of all city residents shop at that grocery store weekly.

Is this inference trustworthy? Choose the best explanation.

A.

Yes — the reporter collected real data from real people, so the inference is valid.

B.

No — the sample only includes people already at that store, so it is not representative of all city residents.

C.

Yes — 87% is a high proportion, so it is probably close to the true value.

D.

No — the reporter needed at least 100 responses for any result to be useful.

$Two dot plots side by side. Plot 1 has dots spread widely from 0.35 to 0.85. Plot 2 has dots clustered tightly from 0.50 to 0.70.$

3.

Two researchers each drew 20 random samples from the same population. Researcher A used samples of size 5; Researcher B used samples of size 20. Their results are shown in the dot plots below.

(a) Which dot plot — Plot 1 or Plot 2 — corresponds to Researcher B's larger samples? Explain how you can tell. (b) Explain how sample size affects the spread of estimates and what that means for the reliability of a single estimate.

4.

A principal wants to estimate what fraction of 900 students at her school bring lunch from home. Which approach gives the MOST reliable estimate?

A.

Survey 8 students she sees in the hallway before school

B.

Randomly select 60 student IDs from the school database and survey those students

C.

Survey all students who raise their hand in homeroom to volunteer

D.

Survey the first 30 students who arrive at the cafeteria each day for a week

E

Error Analysis

1.

Marcus randomly surveyed 30 people at a community center and found that 18 preferred Brand X over Brand Y. He reported his results this way:

"I surveyed 30 people and 18 preferred Brand X. Therefore, 60% of all people prefer Brand X."

What error did Marcus make in his conclusion?

A.

Marcus used the wrong operation — he should have subtracted, not divided.

B.

Marcus stated his sample result as a definite fact about the entire population, without using hedging language.

C.

Marcus's sample size of 30 is too small to draw any conclusion at all.

D.

Marcus should have surveyed people outside the community center, not inside.

2.

Sofia drew one random sample of 40 tiles from a bag and found that 22 were red. She then concluded:

"My sample proportion is $22 \\div 40 = 0.55$ . So the population proportion of red tiles is exactly 0.55."

What error did Sofia make?

A.

Sofia calculated the proportion incorrectly.

B.

Sofia cannot draw any conclusion from just one sample.

C.

Sofia treated one sample's estimate as the exact population value, when it is only an approximation.

D.

Sofia's sample size of 40 is too large and distorts the result.

F

Challenge

1.

A principal wants to know what fraction of her school's 1,200 students walk to school. Design a complete sampling study. Your response must address: (a) the population, (b) a specific random sampling method with a stated sample size, (c) how you would use multiple samples to gauge the variability in your estimate, and (d) how you would state your final inference.

2.