Math problem set Grade 7 Science
Verdon Water District Sampling: Probability and Percent Problems
A grade 7 science problem set where students use a water district's sample results to find probabilities, percents, and predictions, and to reason about sample size.
Objectives
- Students will express a sample result as a fraction, a percent, and a probability.
- Students will predict future results from a sample rate.
- Students will explain why sample size affects how much a result can be trusted.
Vocabulary
Key words: sample, probability, representative, percent
Teacher overview
Water systems test many samples across a year and report how many exceed a safety limit. This problem set uses one fictional system, the Verdon Water District, so students turn its sample counts into fractions, percents, and probabilities, then predict future results and weigh a claim built on very few samples. A common misconception this set corrects is treating the raw count of failing samples as the probability; the count must be divided by the total number of samples first.
Problems
Last year the Verdon Water District collected 200 water samples. Of those, 30 exceeded the safety limit for a contaminant. Use these numbers for the problems below.
- What fraction of the 200 samples exceeded the limit? Write the fraction in simplest form.
- Write that same fraction as a percent.
- An inspector picks one of the 200 samples at random. What is the probability it exceeded the limit? Write it as a decimal.
- What is the probability that a randomly picked sample did not exceed the limit? Write it as a decimal.
- Verdon will collect 40 new samples next year. Using last year's rate, about how many of the 40 would you predict to exceed the limit?
- A nearby system, Halcombe, tested only 10 samples and found 3 over the limit. Halcombe says its water is worse than Verdon's. Find Halcombe's percent over the limit, then explain in one or two sentences why comparing 10 samples to 200 samples can be misleading.
- Verdon splits its 200 samples by zone: the north zone had 24 of 120 samples over the limit, and the south zone had 6 of 80 over. Find the percent over the limit for each zone.
- An inspector picks one sample at random from the north zone's 120 samples. What is the probability it exceeded the limit? Write it as a decimal.
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: 3/20. 30 out of 200 is 30/200, which simplifies to 3/20.
- Answer: 15 percent. 3/20 = 15/100 = 15%.
- Answer: 0.15. 30 / 200 = 0.15, a number between 0 and 1.
- Answer: 0.85. A sample either exceeds the limit or does not, so 1 - 0.15 = 0.85 (also 170/200).
- Answer: 6 samples. At a 15% rate, 0.15 x 40 = 6 samples are predicted to exceed the limit.
- Answer: 30 percent. 3 / 10 = 0.30, or 30%. Even though 30% is higher than Verdon's 15%, 10 samples is a small sample, so a few unusual results swing the percent a lot; 200 samples give a more trustworthy estimate of the true rate.
- Answer: north 20 percent, south 7.5 percent. North: 24 / 120 = 0.20 = 20%. South: 6 / 80 = 0.075 = 7.5%.
- Answer: 0.20. 24 / 120 = 0.20, so the probability is 0.20 (or 20%).
Differentiation
Support
Provide a filled example converting 30 out of 200 into a fraction, a percent, and a decimal, side by side, before problem 1. Allow a calculator for every division and accept a percent written as a decimal where it helps.
Extension
Ask students to design a sampling plan for Verdon next year that would give a more trustworthy zone comparison than this year's, stating how many samples they would take in each zone and why.
ELL supports
Pre-teach "sample," "probability," and "over the limit" with a simple drawing of 200 dots, 30 shaded. Provide the sentence frame "___ out of ___ samples is ___ percent" and allow decimals or fractions as answers.