Water for Schools
Menu

Quiz & test questions Grade 11 Math

Colwyn Flats Rebate Survey: A Grade 11 Sampling Quiz

A 30-point grade 11 quiz on a fictional utility's rebate survey, covering study type, response rate, margin of error, sample size, and nonresponse.

Objectives

  1. Students will classify a data collection as a survey, an experiment, or an observational study.
  2. Students will compute a margin of error and the sample size a stated margin requires.
  3. Students will explain how nonresponse limits what a survey result supports.

Vocabulary

Key words: random sample, response rate, margin of error, nonresponse, population proportion

Teacher overview

This quiz covers survey design, margin of error, sample size, and nonresponse, using a single scenario in which a water department surveys its customers about a proposed rebate for high-efficiency fixtures. It is worth 30 points across eleven items, mixing two multiple-choice items, seven computation items, and two short-answer items that ask students to justify what the sample supports. Students need a calculator and about 40 minutes.

Colwyn Flats Rebate Survey: A Grade 11 Sampling Quiz — Grade 11 Math
Name: ____________________________ Date: ______________

Assessment

The Colwyn Flats Water Department and its rebate survey are fictional. Use these facts for every item.

  • The department serves 8,600 residential accounts.
  • It mailed the survey to 400 accounts chosen at random from its billing list.
  • 312 of those accounts returned a completed survey.
  • Of the accounts that responded, 62 percent said they would apply for the rebate.
  • The department would pay $85 for each rebate approved.
  • Use margin of error = 2 times the square root of (p(1 - p) divided by n), where p is the sample proportion and n is the number of respondents.
  • Use sample size n = 4p(1 - p) divided by the square of the desired margin of error.
  1. (2 points) Which best describes this study? Write the letter.
    A. An experiment, because the department introduced a new rebate.
    B. A sample survey, because responses came from a randomly chosen subset and no treatment was imposed.
    C. A census, because every account had a chance of being selected.
    D. An observational study of metered water use.
  2. (2 points) Find the response rate for the mailed survey, as a percent.
  3. (3 points) Find the margin of error for the 62 percent figure, in percentage points. Round to one decimal place.
  4. (2 points) Using your answer to item 3, give the lower end of the interval around 62 percent. Round to one decimal place.
  5. (3 points) The department wants a margin of error of 3 percentage points instead. How many completed responses would it need? Round up to a whole number.
  6. (3 points) If the response rate stays the same as in item 2, how many surveys must the department mail to collect that many responses? Round up to a whole number.
  7. (3 points) Which is the most serious threat to treating the 62 percent figure as the view of all 8,600 accounts? Write the letter.
    A. The 400 accounts were selected at random.
    B. The 88 selected accounts that did not respond may differ from those that did.
    C. The number of respondents is far above 30.
    D. The result was reported as a percent rather than as a count.
  8. (3 points) Suppose 62 percent of the 8,600 accounts apply for the rebate. What would the department pay in total?
  9. (3 points) Using the lower end of the interval from item 4 instead, how many accounts would apply? Round to a whole account.
  10. (3 points) Explain why the department should report the interval from items 3 and 4 to its board rather than the single figure of 62 percent.
  11. (3 points) A board member proposes surveying only the customers who attended a public meeting about the rebate. Explain why that sample would not support a conclusion about all 8,600 accounts.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key
  1. Answer: B.
    No treatment was assigned to anyone, so the study is not an experiment, and only 400 of the 8,600 accounts were contacted, so it is not a census. A randomly selected subset asked to report an opinion is a sample survey.
  2. Answer: 78 percent.
    312 completed surveys divided by 400 mailed = 0.78, or 78 percent.
  3. Answer: 5.5 percentage points.
    With p = 0.62 and n = 312, the product p(1 - p) = 0.62 x 0.38 = 0.2356. Dividing gives 0.2356 divided by 312 = 0.000755, whose square root is 0.02748. Doubling gives 0.0550, or 5.5 percentage points.
  4. Answer: 56.5 percent.
    62.0 - 5.5 = 56.5 percent at the lower end. The upper end is 67.5 percent.
  5. Answer: 1,048 respondents.
    Using n = 4(0.2356) divided by 0.03 squared gives 0.9424 divided by 0.0009 = 1,047.1, which rounds up to 1,048 completed responses.
  6. Answer: 1,344 surveys.
    At a 78 percent response rate, 1,048 divided by 0.78 = 1,343.6 mailings, which rounds up to 1,344.
  7. Answer: B.
    Nonresponse is the threat here. The 88 accounts that stayed silent may be systematically less interested in the rebate, and nothing in the design corrects for that. Random selection, a large sample, and reporting in percent are all features of a sound survey rather than weaknesses.
  8. Answer: $453,220.
    62 percent of 8,600 accounts is 0.62 x 8,600 = 5,332 accounts, and 5,332 x 85 dollars = 453,220 dollars.
  9. Answer: 4,859 accounts.
    0.565 x 8,600 = 4,859 accounts. Comparing this with item 8 shows the gap between 62 percent and the lower end is worth roughly 473 rebates, or about 40,205 dollars of budget.
  10. Answer: The single figure hides the sampling variability.
    A different random sample of the same size would have produced a somewhat different percentage, and the interval reports how much that figure could reasonably move. A board that budgets from 62 percent alone will treat a range of plausible demand as a single certain number, while the interval from 56.5 percent to 67.5 percent tells the board what the low case and the high case look like before the money is committed.
  11. Answer: Meeting attendees are a self-selected group, not a random sample.
    People who attend a meeting about a rebate are more likely than average to want the rebate, so the group is not representative of the 8,600 accounts. The percentage from such a group would almost certainly run high, and no margin-of-error formula fixes that, because the formula describes only the variability of random selection and says nothing about a biased selection process. The only repair is to draw the sample at random from the whole billing list.

Differentiation

Support

Give students a formula card with both formulas already arranged, with p, n, and the margin of error labeled in words rather than symbols. Allow the use of a worked example on a different proportion, and accept item 10 and item 11 answers given as three bullet points instead of a paragraph.

Extension

Ask students to determine how large the margin of error would have been if only 120 accounts had responded, and to explain why cutting the sample by roughly two-thirds does not triple the margin. A second extension is to compute the sample size needed for a margin of 1 percentage point and to comment on whether the utility would find that survey worth its cost.

ELL supports

Pre-teach "rebate," "response rate," "margin of error," "nonresponse," and "self-selected" with a labeled diagram showing 400 mailed surveys, 312 returned, and 88 unreturned. Provide sentence frames for items 10 and 11, such as "The department should report the interval because ___" and "This group is not representative because ___."

Extensions

Have students find a real customer survey published by a utility, a city, or a state agency, then look for three pieces of information the quiz treats as essential: how respondents were selected, how many were contacted, and how many replied. Reports that omit the response rate are common, and asking students to say what they cannot conclude from such a report is a useful closing exercise for a unit on inference.