Quiz & test questions Grade 12 Math
Amberton County Nitrate Sampling: A Statistical Inference Quiz
A grade 12 quiz on a fictional county's nitrate monitoring, testing sampling-as-inference reasoning and the evaluation of a biased water-quality report.
Objectives
- Students will estimate a population mean nitrate level from a random sample.
- Students will explain why a sample estimate carries uncertainty about the true value.
- Students will identify statistical weaknesses in a biased water-quality report.
Vocabulary
- Nitrate
- Nitrate is a form of nitrogen that plants use as food. It gets into water from fertilizer, animal waste, and septic systems, and too much of it is unhealthy for both people and streams.
Also: random sample, population mean, inference, sampling bias
Teacher overview
This 35-minute quiz (30 points total) checks whether students can treat water sampling as statistical inference and can judge a data-based claim critically. It mixes multiple-choice, short-answer, and computation items built around nitrate monitoring in the fictional Amberton County, a rural system whose aquifer sits beside farmland. For this quiz, the county uses 10 mg/L as its nitrate action limit.
Assessment
Amberton County (a fictional rural water system) drew a random sample of 8 wells and measured nitrate, in mg/L: 6.2, 8.1, 4.5, 9.0, 7.3, 5.8, 10.4, 6.7. Use these readings where a question calls for data.
(2 points) Why does the county test a random sample of wells instead of every tap in the system?
- A) Because testing every tap is against the rules.
- B) Because a random sample lets the county infer the system-wide level without testing everything.
- C) Because all wells in the county are known to be identical.
- D) Because only new wells can legally be tested.
(4 points) Calculate the sample mean nitrate concentration from the 8 readings, in mg/L.
(3 points) What proportion of the sampled wells exceed the 10 mg/L action limit? Give your answer as a percent.
(4 points) Explain what it means to use the sample mean from item 2 as an estimate of the true county-wide mean nitrate level, and state why that estimate is not certain to equal the true value.
(4 points) The county writes that the true average "is probably near 7 mg/L, but could reasonably be somewhat higher or lower." Explain why one sample cannot pin down the exact population mean, and name one change that would narrow the uncertainty.
(5 points) A fertilizer trade group hands out a flyer: "Amberton's water is safe. We tested 3 wells near the town center and all read under 5 mg/L." Identify two statistical weaknesses that make this claim untrustworthy as a county-wide statement.
(4 points) Which sampling plan best supports a trustworthy county-wide nitrate estimate?
- A) Test the three wells known to be cleanest.
- B) Let residents send in samples only when they are worried.
- C) Randomly select wells from across the whole service area.
- D) Test only wells drilled in the last year.
(4 points) Nitrate is a fertilizer farmers apply on purpose, yet it is a contaminant in a drinking-water well. Explain in one or two sentences how the same substance can be a resource on the field and a pollutant in the well.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Answer Key
Answer: B.
A random sample lets the county infer the system-wide nitrate level without the cost of testing every tap. The other options misstate the law or wrongly assume all wells are the same.Answer: 7.25 mg/L.
Sum = 6.2 + 8.1 + 4.5 + 9.0 + 7.3 + 5.8 + 10.4 + 6.7 = 58.0. Mean = 58.0 ÷ 8 = 7.25 mg/L.Answer: 12.5 percent.
One of the 8 wells (10.4 mg/L) exceeds the 10 mg/L limit: 1 ÷ 8 = 0.125, or 12.5 percent of the sampled wells.Answer: the sample mean is a best estimate, not the exact truth.
Using 7.25 mg/L as an estimate means treating the 8 sampled wells as a stand-in for all wells in the county. Because a different random draw of wells would give a slightly different mean, the estimate carries sampling uncertainty and is unlikely to match the true county-wide mean exactly.Answer: one sample varies by chance; a larger random sample narrows it.
Any single sample is just one of many possible draws, so its mean bounces around the true value from sample to sample. Drawing a larger random sample (or repeating the sampling) would shrink that variability and tighten the margin of error around the estimate.Answer: tiny sample and biased location.
First, three wells is far too small to represent a whole county and gives no reliable margin of error. Second, choosing wells near the town center, away from the farmland where nitrate is highest, is a non-random, self-serving location choice that biases the result low. Either weakness alone makes the flyer's safety claim untrustworthy.Answer: C.
Randomly selecting wells from across the whole service area avoids the location and volunteer biases in the other options and best supports a county-wide estimate.Answer: placement decides its role.
On a field, nitrate feeds crops and is a useful nutrient; once it leaches down into groundwater, the same nitrate becomes a contaminant that can make well water unsafe to drink. The substance does not change, only where it ends up.
Differentiation
Support
Provide the eight readings already listed in a column with a blank for the sum, so students can focus on dividing for the mean in item 2. Offer a word bank ("random," "sample," "bias," "margin of error") for the short-answer items.
Extension
Ask students to estimate how the margin of error around the mean would change if the county expanded the sample from 8 wells to 32, and to explain in general terms why larger samples produce tighter estimates without eliminating uncertainty entirely.
ELL supports
Pre-teach "sample," "population," "inference," and "bias" with a picture of a few wells standing in for many. Provide sentence frames for items 4, 5, 6, and 8, such as "The estimate is uncertain because ___" and "The flyer is weak because ___."