7.SP.A.2Common CoreMathStatistics and ProbabilityGrade 7
7.SP.A.2: Using Random Samples to Estimate and Predict
In plain English: 7.SP.A.2 is the Common Core grade 7 math standard that asks students to use data from a random sample to estimate something unknown about a whole population, such as a percent or a mean, or to predict an outcome such as a school election. Students also take many samples of the same size and look at how much the estimates vary, to judge how far off one estimate might be.
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
Common Core State Standards for Mathematics · Domain: Statistics and Probability (SP) · Cluster: Use random sampling to draw inferences about a population. Also written as 7.SP.2 · Official standard
Students learn to answer a question about a large group by studying a small part of it. The population is the whole group you want to know about, such as all 600 students in a school. A sample is the part of the population you actually collect data from. In a random sample, every member of the population has the same chance of being picked, so the sample tends to look like the population (7.SP.A.1). The characteristic of interest is the unknown number you want, such as the percent of students who favor a candidate or the mean (average) length of the words in a book. An inference is a conclusion about the population drawn from sample data, and an estimate is a value that should be close to the true value but is not certain to equal it.
The second half of the standard asks: how far off might the estimate be? Two random samples of the same size almost never give the same result. This natural change from one random sample to the next is called sampling variability. Students gauge it by taking many samples of the same size, or by a simulation (acting out random sampling with a chance tool such as number cubes, a spinner or a random number generator), and looking at how spread out the estimates are on a dot plot (a number line with one dot for each value). The page works through both official examples: estimating the mean word length in a book and predicting a school election. Every data set on this page is invented for teaching.
Learning Objectives
By the end of this lesson, students will be able to:
Name the population, the sample and the characteristic of interest in a statistical question
Use data from a random sample to estimate a percent, a count or a mean for a population
Predict the winner of a school election from randomly sampled survey data
Generate several samples of the same size, real or simulated, and use the spread of their estimates to judge how far off one estimate might be
Explain why estimates from larger random samples vary less than estimates from smaller ones
Prior Knowledge Required
Students should already be comfortable with:
Knowing that a random sample tends to represent the population and that a sample chosen another way can mislead 7.SP.A.1
Making and reading dot plots 6.SP.B.4
Finding the mean of a data set and the range, the largest value minus the smallest 6.SP.B.5
Writing a fraction as a percent and finding a percent of a quantity 6.RP.A.3
Read the prompt aloud and give pairs two minutes to talk before sharing.
Warm-Up Prompt
"A cook makes a big pot of soup. She stirs it well, tastes one spoonful and decides the whole pot needs more salt. Why is one spoonful enough? Why does she stir first? Could a second spoonful taste a little different?"
Collect answers. Students usually say that a well-stirred spoonful is like the whole pot, and that without stirring the spoonful might come from the salty bottom or the watery top. Name the ideas: the pot is the population, the spoonful is a sample, and stirring is like choosing the sample at random. A second spoonful could taste a little different, and that is today's second big idea: samples vary, and we can measure how much.
Direct Instruction20-25 minutes
Part 1: From the sample to the population. For a yes-or-no question, find the fraction of the sample that answered a certain way and write it as a percent. Then assume the population has about the same percent, and find that percent of the population size. For a numerical question, such as word length, find the sample mean (add the values and divide by how many there are) and use it as the estimate of the population mean. Work Examples 1 and 2 below. Stress the words "about" and "estimate": the sample only tells us roughly what the population is like.
Part 2: How far off might it be? Ask: "If the council picked 50 different names, would they get 58% again?" Show Diagram 1, where each dot is the result of one random sample of 50. The estimates form a cluster (a group of dots close together), and the width of the cluster shows how far off one estimate could be. Differences between percents are counted in percentage points: 50% and 58% are 8 percentage points apart. Work Example 3. Point out that we still do not know the true percent for the whole school, but the dot plot tells us which values are believable.
Part 3: Sample size. Show Diagram 2 and work Example 4. The samples of 20 words are all the same size as one another, and so are the samples of 5 words; comparing the two dot plots shows that bigger random samples give estimates that stay closer together. Warn students that a bigger sample only helps if it is still random: 300 answers to an online poll that students choose to answer would still not describe the whole school.
Predicting a school election (official example)
Wilson Middle School has 600 students, and Priya and Marcus are running for student council president. The council has a computer pick 50 names at random from the school list and asks each of those students whom they will vote for (invented data): 29 say Priya and 21 say Marcus.
Equation: Priya's share of the sample is 29 ÷ 50 = 58%. Inference: about 58% of all 600 students favor Priya, which is 0.58 × 600 = 348 students. Prediction: Priya wins, with about 348 of the 600 votes if every student votes
Estimating the mean word length in a book (official example)
A chapter of a class novel has about 2,400 words. Jada uses a random number generator 20 times to pick a page, a line and a word in that line, and counts the letters in each word she lands on (invented data): 2, 2, 2, 4, 3, 9, 7, 3, 3, 6, 3, 6, 2, 6, 8, 3, 4, 7, 4, 9.
Equation: The 20 lengths add to 93 letters, so the sample mean is 93 ÷ 20 = 4.65 letters. Inference: the mean word length in the whole chapter is about 4.7 letters
Gauging how far off a prediction might be
The council repeats the survey from Example 1 twenty times, each time with a new random sample of 50 students (invented data). Diagram 1 shows the percent who favor Priya in each sample.
Equation: The 20 estimates run from 46% to 70%, and their mean is 58.9%. 17 of the 20 are within 8 percentage points of 58%, so one sample of 50 can easily be off by several points. Only 1 sample shows Marcus ahead, so the prediction that Priya wins is fairly safe
Larger samples vary less
Jada's class repeats Example 2 with 15 random samples of 5 words and 15 random samples of 20 words from the same chapter (invented data, means rounded to the nearest tenth). Diagram 2 shows both sets of sample means.
Equation: Samples of 5 words: means from 3.0 to 6.8 letters, a range of 3.8 letters. Samples of 20 words: means from 3.6 to 5.0 letters, a range of 1.4 letters. One sample of 20 words gives a more trustworthy estimate than one sample of 5
Guided Practice15 minutes
A school has 900 students. Eight groups of students each surveyed their own random sample of 25 students, asking "Did you bring lunch from home today?" The table shows how many in each sample said yes (invented data). Work through the questions with the class, one at a time.
Students who brought lunch from home, out of 25, in 8 random samples from a school of 900 (invented data)
Sample
1
2
3
4
5
6
7
8
Yes answers (out of 25)
6
10
9
12
11
8
9
11
Questions and model answers: (1) Write each result as a percent. (24%, 40%, 36%, 48%, 44%, 32%, 36%, 44%; each count is multiplied by 4 because 25 × 4 = 100.) (2) If you only had Sample 1, what would you estimate for the whole school? (24% of 900 = 216 students.) (3) How much do the 8 estimates vary? (From 24% to 48%, a range of 24 percentage points.) (4) Use all 8 samples to make a better estimate. (Together they include 76 yes answers out of 200 students, which is 38%, so about 0.38 × 900 = 342 students.) (5) How far off could one sample of 25 be? (Two of the 8 estimates, 24% and 48%, are 10 or more percentage points from 38%, so one small sample can be far off.) Watch for students who multiply the count by 900 without first making it a percent.
Independent Practice10-15 minutes
Each student works alone. A computer picked 10 of the 240 seventh graders at a school at random, and each one reported how many hours they slept last school night (invented data): 8.5, 7.5, 9, 8, 7, 9.5, 8.5, 8, 7.5, 9 hours. (1) Name the population, the sample and the characteristic of interest. (All 240 seventh graders; the 10 chosen students; the mean hours of sleep on a school night.) (2) Estimate the mean for the population. (82.5 ÷ 10 = 8.25 hours.) (3) Two other random samples of 10 seventh graders had means of 7.8 and 8.7 hours. What does that tell you? (Samples of 10 can differ by almost an hour, so the estimate of 8.25 hours could be off by roughly half an hour.) (4) Name one way to get a more trustworthy estimate. (Use a larger random sample, or combine several random samples.)
Closure5 minutes
Exit ticket: a random sample of 30 of the 450 students at a school found that 12 want school to start later. Estimate how many of the 450 students want a later start. (12 ÷ 30 = 40%, and 40% of 450 = 180 students.) A second random sample of 30 found 15 who want a later start. What does the second sample tell you? (Samples of 30 vary; the true percent may be anywhere from about 40% to 50% or a little outside that, so more samples would help.)
Differentiation Strategies
For Struggling Students
Give a three-step frame for yes-or-no questions: "fraction of the sample, then percent, then that percent of the population"
Use sample sizes of 10, 20, 25 or 50, so that the percents come out as whole numbers
Let students place their sample result on a class dot plot on the board before they write about variation, so they see their dot as one of many
For Advanced Students
Ask students to predict how the dot plot in Diagram 1 would change for samples of 200 students, then test the idea with a random number generator
Give a sample result that suggests a close election (26 of 50) and ask how many more samples they would want before making a prediction, and why
Ask students to find a news poll, name its population and sample size, and explain in a few sentences why the reported result is only an estimate
Assessment Guidance
What to Look For
Check that students name the population and the sample correctly and use the population size, not the sample size, for the final count. In estimates, look for the step from a sample fraction to a percent, then to a count, and for words such as "about" in the conclusion. When students gauge variation, they should point to the spread of several estimates of the same sample size (a range or a cluster on a dot plot), not to a single sample. Listen for the idea that a larger random sample makes the estimates vary less, and that a large sample that is not random still misleads.
02
Classroom Activities
3 Activities
1
Mystery Bag Samples
20 minPairs, then whole class
Each pair samples from a bag of tiles whose makeup they do not know, estimates the percent of red tiles, and then sees how the whole class's estimates spread out.
Materials
One paper bag per pair with 100 tiles or dried beans: 35 red and 65 yellow (do not tell students)
Sticky notes and a class number line on chart paper from 0% to 100%, marked every 10%
Procedure
Without looking, one partner draws 10 tiles, and the other records how many are red
Return all 10 tiles, shake the bag and repeat until the pair has 3 samples of 10
Write each sample's percent of red tiles on a sticky note and place it on the class number line to make a dot plot
Each pair writes an estimate for the whole bag and a sentence about how far off one sample of 10 might be. Then the teacher reveals the true 35%
Sample Class Data (invented, for teacher planning)
Twelve samples of 10 tiles gave these numbers of red tiles: 3, 7, 2, 3, 5, 2, 2, 1, 3, 4, 5, 3. Together that is 40 red tiles out of 120, or about 33%. The single samples ranged from 10% red to 70% red.
Discussion Questions
In the sample data, which sample was farthest from the true 35%, and by how many percentage points? (The sample with 7 red tiles: 70%, which is 35 points too high)
Why is the combined estimate from all the samples closer to 35% than many single samples?
Would samples of 20 tiles spread out more or less on the number line? Why?
Challenge Variation
Pairs repeat the activity with samples of 20 tiles and make a second dot plot under the first, on the same scale. They compare the two dot plots and write a sentence about sample size.
2
Word Length Detectives
20 minPairs
Pairs estimate the mean word length on two facing pages of the class novel, the official example of the standard, from a random sample of 10 words, then compare their estimate with the other pairs' estimates.
Materials
A photocopy of the same two facing pages of the class novel for each pair
A random number generator (calculator or website) or a pair of number cubes
Sticky notes and a class number line from 2 to 8 letters, marked every 0.5
Rules for Counting
Pick a random page (1 or 2), then a random line number, then a random word number in that line. If that word does not exist, pick again
Count letters only: no punctuation. A hyphenated word such as "well-known" counts all its letters (9). Skip numbers written as digits
Collect 10 words, then find the sample mean and round it to the nearest tenth
Procedure
Each pair collects its sample and finds its mean
Pairs place a sticky note with their mean on the class number line, making a dot plot of sample means
Before class, the teacher counts the letters in every word on both pages and finds their true mean, for the reveal at the end
Discussion Questions
How far apart are the smallest and largest sample means on our dot plot? What does that say about one sample of 10 words?
Where would you put a single best estimate, and why?
Why would picking the first 10 words of the chapter not be a random sample?
Modification for Distance Learning
Share one digital page of text. Students use an online random number generator to choose word positions, record the lengths in a shared spreadsheet, and the class builds the dot plot of sample means together on screen.
3
Mascot Poll Simulation
20 minGroups of 3
Groups simulate polls for a mascot vote in which the true support is known, to see how often a poll of 20 or a poll of 40 students predicts the wrong winner.
The Model
Pretend that 45% of the 800 students at a school want the Hawks as the new mascot and 55% want the Owls. A random whole number from 1 to 100 stands for one student: 1 to 45 means Hawks, 46 to 100 means Owls. The Owls should win the real vote.
Procedure
Each group runs 3 polls of 20 students: generate 20 random numbers and count the Hawks votes
Then each group runs 3 polls of 40 students the same way
Record every poll on the board in two rows, "polls of 20" and "polls of 40", and count how many polls show the Hawks ahead
Sample Class Data (invented, for teacher planning)
Ten polls of 20 students gave these numbers of Hawks votes: 10, 7, 9, 13, 5, 8, 7, 11, 9, 13. Ten polls of 40 students gave: 17, 15, 23, 18, 19, 16, 15, 21, 16, 13.
Discussion Questions
How many polls of 20 predicted the wrong winner, with the Hawks ahead? (3 of 10, and 1 more was a tie) How many polls of 40? (2 of 10)
Why does a real poll not know that the true support is 45%? What does the simulation let us see that a real poll cannot?
How large would you want a real poll to be before you trusted it to call a close vote?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Twenty Random Samples of the Same Election
Each dot is one random sample of 50 students from Example 3, placed at the percent of that sample who favor Priya. The estimates cluster between about 52% and 66%. Only one sample (red) falls below the 50% tie line, so the samples agree that Priya is likely to win, even though no single sample gives her exact share.
Diagram 2: Sample Size and Variation
Two dot plots on the same scale, from Example 4. Each dot is the mean word length of one random sample from the same chapter. Means from samples of 5 words spread out over 3.8 letters; means from samples of 20 words stay within 1.4 letters. The Example 2 sample (4.65, rounded to 4.7) is one of the dots in the bottom plot.
04
Homework Assignment
~30 min
7.SP.A.2 Homework: Estimates from Random Samples
Directions: Show your work. For every estimate, name the population and write your answer as a sentence with the word "about". All data are invented.
Part 1: Estimating from One Random Sample (Problems 1-3)
A school has 720 students. A computer picks 40 of them at random, and 14 say they would rather have tacos than pizza for Friday lunch. (a) What percent of the sample prefers tacos? (b) Estimate how many of the 720 students prefer tacos. (c) Why would asking the first 40 students in the lunch line on taco day be a poor way to choose the sample?
The school library has 3,000 books. A student picks 10 books at random and records the number of pages in each: 212, 148, 96, 320, 184, 256, 132, 408, 176, 238. (a) Estimate the mean number of pages for all the books in the library. (b) Another student picks a different random sample of 10 books and gets a mean of 241 pages. Does that mean one of the students made a mistake? Explain.
A factory made 9,000 phone chargers in one day. An inspector tests a random sample of 300 of them and finds 12 that do not work. (a) What percent of the sample does not work? (b) Estimate how many of the 9,000 chargers do not work.
Part 2: Gauging Variation with Many Samples (Problems 4-6)
A middle school has 700 students. Ten groups each surveyed a random sample of 20 students and asked, "Do you play a sport outside school?" The numbers of yes answers were: 12, 9, 12, 11, 7, 10, 10, 10, 11, 10. (a) Write each result as a percent. (b) What is the range of the 10 estimates? (c) Combine all 10 samples to estimate the percent of the school that plays a sport outside school, and the number of students. (d) About how far off could one sample of 20 be?
Jordan and Kim are running for class president at a school of 520 students. Three random samples of 40 students each found 22, 19 and 24 students who plan to vote for Jordan. (a) Write each sample result as a percent. (b) Can you confidently predict the winner? Explain using all three samples. (c) What could you do to make a more trustworthy prediction?
Two groups estimated the mean number of hours of screen time a student at their school has on a school day. Group A took 8 random samples of 5 students; the sample means were 2.3, 3.2, 1.9, 2.1, 2.6, 4.3, 2.7, 2.3 hours. Group B took 8 random samples of 25 students; the sample means were 3.1, 3.1, 2.9, 3, 2.9, 3.2, 2.3, 3.1 hours. (a) Find the range of each group's sample means. (b) Which group's single sample would you trust more? Explain why, using sample size.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Population and Sample
Population, sample and characteristic named correctly
One of the three missing or confused
Population and sample confused
Estimates
Percents, counts and means correct, stated as estimates
One calculation error, or the estimate stated as exact
Several errors or no work
Variation
Uses the spread of several samples of the same size to say how far off one estimate might be
Mentions that samples vary without using the numbers
Treats one sample as the exact answer
Reasoning
Explains the role of random choice and of sample size
Explains one of the two
No explanation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again. All data in the quiz are invented.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A town wants to know how many of its 8,000 households recycle. It surveys 200 households chosen at random. What is the population?
Answer: B
The population is the whole group the town wants to learn about: all 8,000 households. Choice A is the sample, the part that was surveyed. Choice C is the group being counted, not the whole population. Choice D is not part of the question at all.
Question 2 of 20 · Multiple Choice
A student council wants to know what all 640 students at its school think about a new dress code. Which sample is a random sample?
Answer: D
Only a computer pick from the full school list gives every student the same chance of being chosen. Choice A favors students who arrive early. Choice B lets students choose themselves, so only those who care a lot may answer. Choice C leaves out every student who is not on a team.
Question 3 of 20 · Multiple Choice
A school has 560 students. In a random sample of 40 students, 10 ride a bike to school. What is the best estimate of the number of students at the school who ride a bike?
Answer: C
10 ÷ 40 = 25%, and 25% of 560 = 140 students. Choice A divides 560 by 10, the sample count. Choice B is the number who do not bike, 560 - 140. Choice D is only the count in the sample, not an estimate for the school.
Question 4 of 20 · Multiple Choice
An orchard picks 8 apples at random from its harvest and weighs them, in grams: 182, 165, 190, 174, 201, 168, 185, 177. What is the best estimate of the mean mass of all the apples in the harvest?
Answer: A
The sample mean is the estimate: the masses add to 1,442 grams, and 1,442 ÷ 8 = 180.25 grams. Choice B is the median of the sample, the middle value (177 + 182) ÷ 2. Choice C is the sum before dividing. Choice D divides by 7 instead of 8.
Question 5 of 20 · Multiple Choice
A school of 640 students is choosing between two candidates. In a random sample of 80 students, 36 plan to vote for Lena and 44 plan to vote for Omar. About how many votes should Omar expect from the whole school?
Answer: B
Omar has 44 ÷ 80 = 55% of the sample, and 55% of 640 = 352 votes. Choice A is the estimate for Lena, 45% of 640. Choice C is Omar's count in the sample only. Choice D is half of the school, which ignores the survey.
Question 6 of 20 · Multiple Choice
Why would you take several random samples of the same size instead of just one?
Answer: A
Several samples of the same size show the sampling variability: how much the estimates change, which tells you how far off one estimate might be. Choice B is false, because random samples almost never agree exactly. Choice C is false, because samples give estimates, not exact values. Choice D is false, because each sample still has to be random.
Question 7 of 20 · Multiple Choice
Ten random samples of 40 students each were asked whether the school should have a longer lunch. The percents who said yes were 67.5, 55, 45, 60, 67.5, 70, 65, 62.5, 60, 70. Which statement is best supported by these samples?
Answer: C
Nine of the 10 estimates are from 55% to 70%, and their mean is 62.25%, so the true percent is probably somewhere in that range, but no one sample tells us exactly where. Choice A treats one sample value as exact. Choice B is supported by only 1 of the 10 samples (45%). Choice D ignores that the estimates cluster together.
Question 8 of 20 · Multiple Choice
A class took 7 random samples of 15 students and found the mean hours of sleep in each: 8, 7.9, 7.7, 8.3, 8.2, 7.9, 8.4 hours. What is the range of these sample means?
Answer: A
The range is the largest value minus the smallest: 8.4 - 7.7 = 0.7 hours. Choice B is only the largest sample mean. Choice C is only the smallest. Choice D is about the mean of the seven sample means (56.4 ÷ 7), which is a center, not a spread.
Question 9 of 20 · Multiple Choice
Four classes each took 20 random samples to estimate the percent of students who walk to school. Which class should get estimates that vary the least?
Answer: D
Larger random samples give estimates that stay closer together, so samples of 100 vary the least. Choices A, B and C are all smaller samples; samples of 10 vary the most, because one or two unusual students change the percent a lot.
Question 10 of 20 · Multiple Choice
A random sample of 12 words from a magazine article has these lengths, in letters: 4, 7, 2, 5, 9, 3, 6, 4, 8, 3, 5, 4. What is the best estimate of the mean word length in the article?
Answer: D
The lengths add to 60 letters, and 60 ÷ 12 = 5 letters. Choice A is the median, (4 + 5) ÷ 2. Choice B is the mode, the length that appears most often. Choice C is the sum before dividing.
Question 11 of 20 · Multiple Choice
A computer picked 60 seventh graders at random from all the seventh graders at Lincoln Middle School. 45 of them have a phone. Which conclusion is best supported?
Answer: B
The sample was chosen at random from Lincoln seventh graders, so it supports an estimate for that population: 45 ÷ 60 = 75%. Choice A goes beyond the population that was sampled. Choice C confuses the sample count with the population count. Choice D includes grades 6 and 8, which were not sampled.
Question 12 of 20 · Multiple Choice
Six random samples of 100 voters in a town each gave these percents who will vote yes on a new park: 47, 54, 54, 55, 53, 46. Based on these samples, about how far from the true percent could one sample of 100 be?
Answer: B
The estimates spread from 46% to 55%, so they sit within 4.5 points of the middle of that spread, 50.5%. The true percent is probably inside the cluster, so one sample can be off by about 5 percentage points. Choice A is too small: all 6 samples are more than 1 point from 50.5%. Choices C and D are far larger than any difference in the data.
Question 13 of 20 · Multiple Choice
A forest has 2,000 trees. In a random sample of 40 trees, 6 have a leaf disease. What is the best estimate of the number of trees in the forest with the disease?
Answer: D
6 ÷ 40 = 15%, and 15% of 2,000 = 300 trees. Choice A divides 2,000 by 6. Choice B is the number of healthy trees, 2,000 - 300. Choice C is the count in the sample only.
Question 14 of 20 · Multiple Choice
Which plan is the best way to gauge how far off an estimate from one random sample of 30 students might be?
Answer: B
Many samples of the same size, each chosen at random, show how much estimates vary. Choice A repeats the same students, so it measures whether they change their minds, not sampling variability. Choice C samples a different population. Choice D does not produce any new data about the population.
Question 15 of 20 · Short Answer
Ana and Ben are running for class president. Three random samples of 50 students found 27, 24 and 26 students who plan to vote for Ana. Write each result as a percent. Can you confidently predict the winner? Explain.
The results are 54%, 48% and 52%. Two samples show Ana ahead, and one shows Ben ahead. Together the samples have 77 Ana votes out of 150, about 51%, which is very close to 50%. Ana may be slightly ahead, but the race is too close to predict with confidence: samples of 50 vary by several percentage points. Larger samples, or more of them, would help.
Question 16 of 20 · Short Answer
A student wants to know how many hours of sports practice students at her school have each week. She surveys 25 students at the gym after soccer practice. Explain why her estimate may be far off, and describe a better way to choose the sample.
Students at the gym after practice play a sport, so the sample leaves out students who do not practice, and the estimate will be too high. The sample is not random. A better plan: get the list of all students, number the names and use a random number generator to pick 25, so every student has the same chance of being chosen.
Question 17 of 20 · Short Answer
Describe a plan for estimating the mean word length in a 180-page book without counting every word. Say how you will choose the words, how many you will use, and how you would check how far off your estimate might be.
Sample answer: use a random number generator to pick a page from 1 to 180, then a line and a word in that line; repeat to get 30 random words and count the letters in each (letters only). The sample mean is the estimate of the mean word length. To gauge how far off it might be, take several more random samples of 30 words, make a dot plot of their means, and look at how spread out they are. Any plan with random word choice, a stated sample size and repeated samples of the same size earns full credit.
Question 18 of 20 · Short Answer
A school has 840 students. In a random sample of 60 students, 21 have a public library card. Estimate the number of students at the school who have a library card. Show your work.
21 ÷ 60 = 35% of the sample. 35% of 840 = 0.35 × 840 = 294 students. So about 294 of the 840 students have a library card. The answer is an estimate, because a different random sample would give a slightly different number.
Question 19 of 20 · Short Answer
Class A took 20 random samples of 10 students and Class B took 20 random samples of 40 students, each to estimate the percent of students who like the new cafeteria menu. The estimates from Class A spread from 20% to 80%. The estimates from Class B spread from 40% to 62%. Which class's single sample would you trust more? Explain.
Class B. Its estimates spread over 62 - 40 = 22 percentage points, while Class A's spread over 80 - 20 = 60 points. Samples of 40 are larger, so one or two unusual students change the percent less, and the estimates stay closer together. A single sample of 40 is likely to be much closer to the true percent than a single sample of 10.
Question 20 of 20 · Short Answer
A cereal factory fills boxes that should hold 400 grams. Five random samples of 10 boxes had these mean masses, in grams: 402.9, 402.4, 401.8, 403.6, 403.1. Estimate the mean mass of all the boxes and say about how far off one sample mean might be.
The mean of the five sample means is 2,013.8 ÷ 5 = about 402.8 grams, so the boxes hold a little more than 400 grams on average. The sample means run from 401.8 to 403.6 grams, a range of 1.8 grams, so one sample mean might be off by about 1 gram.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 7.SP.A.2 mean?
7.SP.A.2 means students can use a random sample to make an estimate or a prediction about a whole population, and can judge how far off it might be. For example, a survey of 50 randomly chosen students can estimate how the whole school will vote. Students then compare many samples of the same size to see how much the estimates change.
Is 7.SP.A.2 taught in grade 7 or grade 8?
It is a grade 7 standard. It comes right after 7.SP.A.1, where students learn why a sample should be random, and before 7.SP.B.4, where they use samples to compare two populations. In high school, students build on it with margins of error (HSS.IC.B.4).
What is the difference between a population and a sample?
The population is the whole group you want to know about, and the sample is the part of it you actually collect data from. If a school wants to know about all 600 of its students and surveys 50 of them, the 600 students are the population and the 50 are the sample.
Why does the sample have to be random?
A random sample gives every member of the population the same chance of being chosen, so it tends to look like the population. A sample chosen for convenience, such as the students in one club, can differ from the population in ways that make the estimate wrong, no matter how large the sample is.
How do you estimate a number in the population from a sample?
Find the percent of the sample with the characteristic, then take that percent of the population size. If 12 of 40 sampled students walk to school, that is 30%, and 30% of a school of 700 is about 210 students. For a mean, use the sample mean as the estimate.
What does "gauge the variation in estimates" mean for a grade 7 student?
It means taking several random samples of the same size, finding the estimate from each, and looking at how spread out the estimates are. If most estimates fall within a few percentage points of each other, one estimate is probably not far off. If they spread over 30 points, one estimate could be far off.
Do students need to calculate a margin of error for 7.SP.A.2?
No. Grade 7 students judge variation informally, from a dot plot or the range of several estimates. The formal margin of error is a high school topic (HSS.IC.B.4). Words such as "about", "probably" and "give or take a few points" are the right level.
Why do larger samples give better estimates?
In a larger random sample, one or two unusual members change the result less, so estimates from repeated samples stay closer together. Diagram 2 shows this: sample means from 5 words spread much more than sample means from 20 words. A larger sample only helps if it is still random.
What is a simulation, and how is it used here?
A simulation acts out random sampling with a chance tool, such as a random number generator, number cubes or a spinner, instead of surveying real people. When the true value is built into the model, as in the Mascot Poll Simulation, students can see how often a sample points to the wrong answer.
What mistakes do students make with 7.SP.A.2?
A frequent mistake is to treat one sample result as exact, for example saying "exactly 58% will vote for Priya". Other errors are using the sample size instead of the population size in the final step, trusting a large sample that was not random, and comparing estimates from samples of different sizes as if they should vary the same amount.
07
Related Standards
6 standards
These standards connect to 7.SP.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
7.SP.A.1Prerequisite
Understand that a representative, random sample supports valid inferences about a population
