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Statistics math worksheets

Build your own math worksheets from 35,000+ problems for grades 3 to 12, from fractions to AP Calculus. Every problem comes with step-by-step solutions.

Learning from data

Learning goals

  • Identifies the context, cases, and variables in a data investigation.
  • Frames questions that can be answered with data.
  • Distinguishes descriptive patterns from inferential claims.
  • Uses context to guide appropriate summaries and conclusions.

Click problems to add them to your worksheet.

Learning from data 1
A wildlife center has records for \(642\) rescued birds. A researcher selects \(50\) records and notes the recovery time for each bird. In this context, what does one datum represent, and what is the data set?

Hints

- Identify what is recorded for one selected bird. - Then describe the collection formed by repeating that recording for every selected bird.

Solution

1. One datum is the recorded recovery time for one selected bird. 2. The data set is the collection of the \(50\) recorded recovery times.

Answer

One datum is one bird’s recovery time; the data set is all \(50\) recovery times.
Learning from data 2
A district contains \(N=12{,}480\) high school students. A survey is completed by \(n=625\) of them. Interpret \(N\) and \(n\) in context.

Hints

- Connect each symbol to either the full group or the observed subset. - Check which symbol denotes the size of each group rather than a measured variable.

Solution

1. \(N=12{,}480\) is the population size: all high school students in the district. 2. \(n=625\) is the sample size: the students whose survey responses were collected.

Answer

\(N=12{,}480\) is the population size, and \(n=625\) is the sample size.
Learning from data 3
A factory produced \(24{,}000\) ceramic tiles in one week. Inspectors test \(160\) tiles. Which statement correctly describes the sample? a) Every tile produced that week b) The \(160\) tested tiles c) The measurements recorded on the tested tiles Explain.

Hints

- Separate the selected items from the information recorded about them. - Match each answer choice to population, sample, or data before choosing.

Solution

1. A sample is the subset of items selected from the population. 2. Therefore, the sample is the \(160\) tested tiles; the measurements are the data obtained from them.

Answer

b) The \(160\) tested tiles are the sample.
Learning from data 4
A study of community gardens records the number of pounds of produce harvested by each of \(35\) selected plots. Give an example of one datum and describe the complete data set without inventing values.

Hints

- Describe the form of the information rather than making up a numerical observation. - Tie one datum to one observational unit and include the measurement unit.

Solution

1. One datum is the harvest weight, in pounds, from one selected plot. 2. The data set is the collection of all \(35\) recorded harvest weights.

Answer

One datum is one selected plot’s harvest weight; the data set is all \(35\) such weights.
Learning from data 5
A city reports that the mean age of all firefighters employed by the city is \(38.6\) years. Is \(38.6\) a statistic or a parameter? Explain.

Hints

- Identify whether the numerical summary comes from a sample or the whole population. - A parameter describes an entire defined population.

Solution

1. The mean describes all firefighters employed by the city, which is the population of interest. 2. Therefore, \(38.6\) years is a parameter.

Answer

It is a parameter because it summarizes the entire population of city firefighters.
Learning from data 6
A sculpture park has exactly \(73\) permanent sculptures. Staff record the material of every permanent sculpture before updating the park guide. Is this data collection a sample or a census? Identify the population and justify your classification.

Hints

- Compare the observed group with the full group named in the study. - Decide whether any population members were left unobserved. - Use the classification that applies when the entire population is recorded.

Solution

1. The population is all \(73\) permanent sculptures in the park. 2. Staff record information from every member of that population. 3. Therefore, the data collection is a census.

Answer

It is a census. The population is all \(73\) permanent sculptures, and every sculpture in that population is included.
Learning from data 7
A school wants to know how much its buses' arrival times vary. It has a posted schedule and a list of actual arrival times over ten school days. Which would answer that question, and why?

Hints

- Does a scheduled arrival tell you exactly when the bus arrived? - What would the list of real arrival times tell you that the schedule would not?

Solution

1. The posted schedule says when the buses are supposed to arrive, not how the arrivals vary. 2. Actual arrival times from several days show the differences among observations. 3. Use the list of actual times to study variation.

Answer

Use the actual arrival times from ten days. A posted schedule does not show how much the times vary.
Learning from data 8
A city park wants to describe how far visitors jog. Which records would help: each visitor's distance or just the total distance jogged by everyone? Explain.

Hints

- Decide what one observation should represent. - Can a single total tell how far each visitor jogged? - Think about what list would allow a distribution of distances.

Solution

1. The question concerns differences among visitors' distances. 2. A list with one distance per visitor shows those differences. 3. One total for all visitors hides the individual distances.

Answer

Record each visitor's jogging distance. The combined total cannot show how individual distances vary.
Learning from data 9
A student measures the heights of the \(12\) players on one basketball team. Which claim describes only these players, and which makes a claim about a larger group? a) “The shortest measured player is \(160\,\text{cm}\) tall.” b) “Basketball players at every school are at least \(160\,\text{cm}\) tall.”

Hints

- Identify exactly whose heights were measured. - Check whether each statement stays within that group. - One observed minimum does not automatically become a minimum for everyone.

Solution

1. Claim a) describes an observed minimum in the measured team. 2. Claim b) makes a statement about players beyond the team. 3. Data from one team alone cannot establish a universal minimum for all schools.

Answer

a) Describes the measured team. b) Claims something about a larger group and is not supported by one team's measurements.
Learning from data 10
A manufacturer produced \(84{,}000\) safety bolts. To measure tensile strength, a testing machine pulls a bolt until it breaks. Explain why a statistical study based on a sample is preferable to testing every bolt.

Hints

- Identify what physically happens to a bolt during the stated measurement. - Compare the consequence of testing the entire production lot with testing only a subset. - Focus on why the measurement process itself makes a census impractical.

Solution

1. The tensile-strength test destroys each bolt that is tested. 2. Testing all \(84{,}000\) bolts would leave no usable bolts and would also require unnecessary time and resources. 3. A suitably selected sample can provide information about the tensile strengths of the full production lot while destroying only a small fraction of the bolts.

Answer

Because the strength test destroys each tested bolt, using a sample provides information about the full lot without destroying every bolt.
Learning from data 11
A museum wants to know how long visitors spend in a new exhibit. Its plan states the population and the sample but does not state what will be recorded from each selected visitor. Identify the missing component needed to answer the question.

Hints

- Ask what piece of information must be collected from each sampled individual. - Name both the characteristic and a unit that would make observations comparable.

Solution

1. The study needs a datum from each selected visitor. 2. The missing information is each visitor’s time spent in the exhibit, measured in a specified unit such as minutes.

Answer

The plan must specify that the time each selected visitor spends in the exhibit will be recorded, such as in minutes.
Learning from data 12
A study uses a sample of \(120\) April receipts from Juniper Grill to investigate tipping behavior there. Nadine says the population is “the \(120\) receipts.” Correct the statement.

Hints

- Distinguish the records actually selected from the larger collection they represent. - The population should use the same place and time boundaries as the sampling frame.

Solution

1. The \(120\) receipts are the sample. 2. The population is all receipts, or all customer transactions represented by receipts, at Juniper Grill during April.

Answer

The \(120\) receipts are the sample; the population is all receipts at Juniper Grill during April.
Learning from data 13
Sabeen wants to study the typical length of songs played by WCRH, Cedar Ridge University's campus radio station. The station has played \(11{,}600\) songs this semester. Explain what must be sampled and what datum should be collected from each sampled unit.

Hints

- Identify the observational units in the population and the characteristic the study wants to summarize. - Keep the sampling unit aligned with the quantity being measured.

Solution

1. A subset of the songs played by WCRH this semester should be sampled. 2. For each sampled song, its length should be recorded in a consistent unit, such as seconds.

Answer

Sample songs played by WCRH this semester and record the duration of each sampled song, such as in seconds.
Learning from data 14
Put these components in a logical order for planning a statistical study: analyze the data, define the investigative question, collect data from a sample, identify the population.

Hints

- Think about what must be decided before any observations can be collected. - Arrange the components so that each later step uses decisions made earlier.

Solution

1. Define the investigative question. 2. Identify the population addressed by that question. 3. Collect data from a sample of the population. 4. Analyze the collected data.

Answer

Define the investigative question; identify the population; collect data from a sample; analyze the data.
Learning from data 15
In a sample of \(250\) registered voters, \(58\%\) support a proposed bond. The true percentage among all registered voters in the county is unknown. Identify the statistic and the parameter.

Hints

- Identify whether the numerical summary comes from a sample or the whole population. - Trace the stated percentage back to the group from which it was calculated.

Solution

1. The sample percentage, \(58\%\), is a statistic. 2. The unknown percentage among all registered voters in the county is the population parameter.

Answer

Statistic: \(58\%\). Parameter: the true countywide percentage of registered voters who support the bond.
Learning from data 16
A sample of \(40\) light bulbs has a mean lifetime of \(1184\,\text{hours}\). The manufacturer wants to estimate the mean lifetime of every bulb of that model. Identify the statistic, parameter, and observational units.

Hints

- Match each term to the sample, the population, or the items measured. - The individual bulb is measured, while the mean summarizes a group of bulbs.

Solution

1. The statistic is the sample mean, \(1184\,\text{hours}\). 2. The parameter is the unknown mean lifetime of all bulbs of that model. 3. The observational units are the individual sampled light bulbs.

Answer

Statistic: \(1184\,\text{hours}\). Parameter: the population mean lifetime. Observational units: individual sampled bulbs.
Learning from data 17
A retailer samples \(300\) of its online orders from June and finds that \(4.7\%\) were returned. An employee says, “The parameter is \(4.7\%\).” Explain the error.

Hints

- Trace where the numerical value came from. - The unknown population percentage is the parameter the sample statistic is intended to estimate.

Solution

1. The value \(4.7\%\) was calculated from the sample, so it is a statistic. 2. The parameter is the unknown percentage of all the retailer’s June online orders that were returned.

Answer

\(4.7\%\) is a statistic; the parameter is the unknown return percentage for all the retailer’s June online orders.
Learning from data 18
An online archive studies how many document pages are viewed during a visit. Its sample contains data from \(900\) website sessions created by \(520\) distinct visitors. For this investigative question, identify the observational units and the sample size. Explain why the number of distinct visitors is not the sample size.

Hints

- Match one recorded value to the item or event that produced it. - Decide whether the question concerns people or separate visits. - Count observational units rather than unique identities.

Solution

1. The variable is pages viewed during one website session, so each session is an observational unit. 2. The sample contains \(900\) sessions, giving sample size \(n=900\). 3. Some visitors created more than one session, so the \(520\) distinct people do not equal the number of observed visits.

Answer

The observational units are website sessions, and the sample size is \(n=900\). The \(520\) visitors are not the sample size because a visitor may contribute more than one session.
Learning from data 19
A county has \(96\) public playgrounds. The mean age of all \(96\) playgrounds is \(11.4\) years. A random sample of \(24\) playgrounds also happens to have mean age \(11.4\) years. Identify the parameter and the statistic, and explain why they are different even though their numerical values are equal.

Hints

- Trace each mean back to the group from which it was calculated. - Classify the summaries by their source rather than by their numerical size. - Consider whether equal numbers must represent the same statistical role.

Solution

1. The population mean age of all \(96\) playgrounds is the parameter. 2. The sample mean age of the \(24\) selected playgrounds is the statistic. 3. A parameter and a statistic are distinguished by the group summarized, not by whether their numerical values happen to match.

Answer

Parameter: the mean age of all \(96\) playgrounds, \(11.4\) years. Statistic: the mean age of the \(24\) sampled playgrounds, also \(11.4\) years. They differ because one summarizes the population and the other summarizes a sample.
Learning from data 20
A Pinebrook Clinic data table has one row for each of \(75\) patients and six recorded variables in each row. Nayra says the sample size is \(450\) because the table contains \(75\cdot6=450\) entries. Correct Nayra's reasoning.

Hints

- Determine what one row represents. - Separate the number of observational units from the number of variables recorded per unit. - Recall what the symbol \(n\) counts.

Solution

1. Sample size counts observational units, not the total number of recorded cells. 2. Each row represents one patient, so there are \(75\) observational units. 3. The six variables provide multiple data values about each patient but do not create additional patients.

Answer

The sample size is \(n=75\), not \(450\). The table contains six variables for each of \(75\) patient observational units.
Learning from data 21
A museum records the time visitors spend in its new gallery. A curator asks, “Do visitors spend more time in the new gallery than in the permanent gallery?” The available file contains times only for the new gallery. Explain why the file cannot answer the comparative question and identify the missing data.

Hints

- Identify every group named in the question. - Check whether the file contains observations for each group. - State what additional distribution is needed for the comparison.

Solution

1. The question compares two gallery-time distributions. 2. The file supplies observations for only the new gallery, so there is no reference distribution for the permanent gallery. 3. Comparable visit-time data for the permanent gallery are also required.

Answer

The file cannot answer the question because it includes only one comparison group. Visit-time data for the permanent gallery are missing.
Learning from data 22
A delivery service records pickup and drop-off times in several cities. Some entries use local time; others use UTC (a single worldwide time standard). The file does not say which clock each entry uses. Why can't the service subtract these times yet? What information does it need?

Hints

- A clock time alone may not tell you the actual moment of an event. - Think about the same clock reading in two different cities. - What information is needed before comparing or subtracting the times?

Solution

1. Pickup and drop-off times must use the same time zone before subtracting them. 2. The same clock reading can represent different moments in different time zones. 3. Identify the time zone for each recorded time and convert all times to one common time zone.

Answer

Subtracting mixed clock times can give incorrect trip lengths. Identify their time zones and put all times on one common clock first.
Learning from data 23
A help desk writes “90+” instead of an exact number whenever a caller waits more than \(90\) minutes. Can these records give the exact longest wait or the exact mean wait? Explain what is known about the “90+” waits.

Hints

- Does “90+” record one number or a range of possible numbers? - Which actual wait times would you need to find the maximum? - Which times would you need to add when calculating the mean?

Solution

1. Each “90+” entry tells us only that the wait was longer than \(90\) minutes. 2. The longest wait could be any of those unknown larger values, so its exact value is unknown. 3. The exact mean also needs every wait time, including the ones shown only as “90+.”

Answer

No. Each “90+” wait exceeded \(90\) minutes, but its exact length is unknown. Neither the exact maximum nor the exact mean can be found.
Learning from data 24
A library's checkout list has \(1260\) rows but only \(900\) different loans. Some books have several authors, so those loans appear once for each author. The library wants to study how long a loan lasts. Why would using all \(1260\) rows give a misleading distribution? How many observations should be used?

Hints

- What is the individual thing whose duration the library wants to describe? - Why does a book with several authors show up more often in this list? - How many different loans are there after removing repeated entries?

Solution

1. One observation should represent one library loan, not one author on a loan. 2. The longer list counts some loans more than once and gives them too much weight. 3. Use one checkout-duration value per distinct loan, giving \(900\) observations.

Answer

Some loans would be counted several times. Use one duration per loan, for \(900\) observations.
Learning from data 25
A rail operator asks, “What is the distribution of arrival delays for trains on Route K?” The available schedule lists only planned arrival times, not actual arrival times. Explain why the schedule cannot answer the question and identify the required data for each train trip.

Hints

- Define what must be compared to obtain one delay. - Check which part of that comparison is absent. - State the complete datum needed for each trip.

Solution

1. A delay compares an actual arrival time with the corresponding planned arrival time. 2. The schedule supplies only the planned times, so no trip-level delay can be calculated. 3. Each train trip needs both its planned and actual arrival times, or a directly recorded arrival delay.

Answer

The schedule alone cannot provide delays. Each Route K trip needs a matched planned and actual arrival time, or a recorded delay value.
Learning from data 26
A neighborhood library wants to study how long visitors use its public computers. Name the observational unit, the numerical variable, and a suitable measurement unit. Explain what to record for each visit.

Hints

- Identify which individual event gets one data record. - Distinguish a duration from a clock time. - Choose a consistent unit so values can be compared.

Solution

1. The observational unit is one computer-use visit. 2. The variable is elapsed time using a computer. 3. Record the length of each visit in minutes, not just its start time or a daily total.

Answer

Unit: one computer-use visit. Variable: duration of that visit, measured in minutes. Record one duration per visit.
Learning from data 27
Two ticket windows at a movie theater record waiting times. The lobby window has waits of \(2,3,3,7\) minutes; the outdoor window has waits of \(3,3,4,5\) minutes. Compare their means and longest waits. Could the theater tell which window had the longer maximum wait from the means alone? Explain.

Hints

- Compare each set's total and its number of waits. - Identify the largest value in each set. - Can two sets with one equal summary still differ in another feature?

Solution

1. Each set totals \(15\) minutes for four waits, so each mean is \(3.75\) minutes. 2. The longest waits are \(7\) minutes at the lobby window and \(5\) minutes at the outdoor window. 3. Since the means are equal but the maxima differ, a mean alone cannot identify the longest wait.

Answer

Both means are \(3.75\) minutes. The longest waits are \(7\) minutes (lobby) and \(5\) minutes (outdoor). No; the means are equal, but the longest waits differ.
Learning from data 28
A transit agency records the number of riders in each train car whenever the train reaches a station. An analyst wants to answer, “What is the typical length of an individual rider’s trip?” Explain why the recorded counts are not sufficient, and identify the additional information needed.

Hints

- Identify the observational unit required by the question. - Ask whether the current records connect the beginning and end of the same observation. - State what one complete datum would need to contain.

Solution

1. The car counts show how many riders are present at each station, but they do not link a rider’s boarding station to that rider’s exit station. 2. Trip length is defined for an individual ride, so the data must identify where each observed ride begins and ends, or directly record its distance or number of stops.

Answer

The counts are not sufficient because they do not track individual rides from boarding to exit. The agency needs each ride’s starting and ending stations, or an equivalent trip-length measurement.
Learning from data 29
A public library wants to estimate the proportion of all attempts to borrow a digital book that fail because every license is in use. The library’s current file contains only completed digital checkouts. Can that file answer the investigative question? Explain and identify what records are missing.

Hints

- Compare the population named in the question with the observations included in the file. - Determine whether both possible outcomes are represented. - Identify the denominator required for the requested proportion.

Solution

1. The population of interest is all attempts to borrow a digital book, including successful and unsuccessful attempts. 2. The current file contains only successful attempts, so failures are absent and the desired proportion cannot be calculated. 3. The library must record every borrowing attempt and whether it resulted in a completed checkout or a license-unavailable failure.

Answer

No. The file excludes failed attempts, so it cannot provide the requested proportion. Records of all borrowing attempts and their outcomes are needed.
Learning from data 30
A college wants to study the number of credits carried by students enrolled on September \(15\). Its file includes everyone who registered at any time during the semester, including students who withdrew before September \(15\) and students who enrolled afterward. Explain why the file does not match the population in the investigative question, and state which records should be included.

Hints

- Locate the time condition that defines membership in the population. - Compare that condition with the rule used to build the file. - Decide which records fall outside the stated boundary.

Solution

1. The population is defined by enrollment status on September \(15\), not by whether a person registered at any point during the semester. 2. Students who withdrew before that date and students who enrolled later are outside the population. 3. The file should contain only students enrolled on September \(15\), with each student’s credit load on that date.

Answer

The file uses the wrong time boundary. It should include exactly the students enrolled on September \(15\) and record each one’s number of credits on that date.
Learning from data 31
A greenhouse sensor records temperature every \(10\) minutes. A researcher’s question is, “What was the highest temperature on each day in July?” The raw file has one row for every sensor reading. For the final data set that answers the question, what should the observational units and data values be?

Hints

- Identify the phrase that tells how many final values the question requires. - Distinguish raw sensor readings from the summarized observations used in the analysis. - Decide what one completed row of the analysis data set should represent.

Solution

1. The question asks for one result for each day, so the observational units in the final data set are the days in July. 2. All readings within a day must be combined to find that day’s maximum temperature. 3. Each datum in the final data set is the highest recorded temperature for one day.

Answer

The observational units should be the days in July, and each datum should be that day’s maximum recorded temperature.
Learning from data 32
A marine center asks, “What is the typical recovery time for all sea turtles treated this year?” Its data file contains recovery times only for green sea turtles, although the center also treated loggerhead and leatherback turtles. Explain why the file cannot answer the stated question and give two valid ways to resolve the mismatch.

Hints

- Compare the group named in the question with the group represented in the file. - Look for members of the population who have no data. - Consider changing either the data collection or the scope of the question.

Solution

1. The stated population includes all sea turtles treated this year, but the file represents only green sea turtles. 2. One resolution is to collect recovery times for every treated species so the data match the original population. 3. Another resolution is to narrow the investigative question to green sea turtles treated this year.

Answer

The file does not represent the full stated population. The center can either add data for the other treated species or revise the question so it concerns only green sea turtles treated this year.
Learning from data 33
A phone help line wants to know how much individual callers' waiting times vary. Its monthly report gives one average wait for each of \(18\) offices. Why can't those averages show the spread of individual waits?

Hints

- Decide whether the question is about offices or individual calls. - What information is lost when many individual times become a single average? - Think about two offices with the same average but very different waits.

Solution

1. Each reported value is an office average, not one caller's wait. 2. Offices with the same average could have very different spreads among their callers. 3. To describe individual waiting times, the help line needs the wait time recorded for each sampled call.

Answer

Office averages hide differences among individual callers. The individual wait times are needed to describe their spread.
Learning from data 34
Two ovens bake the same type of ceramic piece. A technician records the firing temperature for each piece but accidentally removes the oven label from every record. Can the resulting data answer, “How do firing-temperature distributions compare between Oven A and Oven B?” Explain.

Hints

- List the pieces of information needed to form each comparison group. - Determine whether every recorded value can still be linked to its source. - Distinguish analyzing pooled data from comparing labeled groups.

Solution

1. The temperature values remain available, but the contextual variable identifying the oven is missing. 2. Without that label, each observation cannot be assigned to Oven A or Oven B. 3. The combined values can describe overall firing temperatures, but they cannot support a comparison between the two ovens.

Answer

No. The oven label is required to separate the observations into the two groups. Without it, only the combined temperature distribution can be studied.
Learning from data 35
An outdoor festival wants to estimate the proportion of attendees who used public transportation. Its ticket file has one row for each online purchaser, but a purchaser may buy tickets for several people. Explain why purchasers are not the correct observational units for the stated question and identify the appropriate units.

Hints

- Match the denominator of the requested proportion to the units in the data. - Check whether one database row can stand for more than one person in the population. - Identify who actually has the characteristic being measured.

Solution

1. The question concerns individual attendees, not ticket transactions or purchasers. 2. One purchaser can represent several attendees whose transportation choices may differ. 3. The observational units should be individual festival attendees, with transportation method recorded for each attendee.

Answer

Purchasers are not the correct units because one purchase can cover several attendees. The observational units should be individual attendees.
Learning from data 36
A sample contains \(240\) streetlights. Inspection results are missing for \(18\) of them, and \(40\) of the \(222\) streetlights with recorded results failed. Calculate the recorded-result completion rate and the failure rate among streetlights with known results. Explain why a report should state both denominators.

Hints

- Separate the question about available results from the question about failures. - Identify the appropriate denominator for each percentage. - Do not silently treat missing outcomes as successful inspections.

Solution

1. The recorded-result completion rate is \(\frac{222}{240}=0.925=92.5\%\). 2. The failure rate among known results is \(\frac{40}{222}\approx0.1802\approx18.0\%\). 3. The first denominator describes data completeness, while the second describes failures only among streetlights whose outcomes are known.

Answer

The completion rate is \(92.5\%\). The failure rate among known results is approximately \(18.0\%\). Both denominators should be reported because missing inspection outcomes are not known passes or failures.
Learning from data 37
A warehouse wants the proportion of shipped items that were damaged. Its file has one row per order and records only whether the order contained at least one damaged item. Explain why this order-level file cannot determine the item-level damage proportion, and state what additional counts are required.

Hints

- Identify the observational units named in the requested proportion. - Compare those units with what one file row represents. - Determine the numerator and denominator that the desired proportion needs.

Solution

1. The requested proportion uses individual shipped items as observational units. 2. An order may contain several items, and the current indicator does not show how many items were damaged or how many items were shipped. 3. The warehouse needs the total number of shipped items and the number of damaged items, ideally recorded at the item level.

Answer

The file summarizes orders rather than items, so it cannot provide an item-level damage proportion. The total number of shipped items and the number of damaged items are required.
Learning from data 38
A city wants to study the change in annual water use for each municipal building after conservation upgrades. The current file contains only each building’s water use after the upgrades. Explain why the file cannot answer the question and describe the paired information needed.

Hints

- Determine how a change value is defined for one observational unit. - Check whether both required time points are present. - Explain why the measurements must be linked to the same building.

Solution

1. A change for one building requires two measurements for that same building. 2. The current file contains only the after-upgrade value, so a before-to-after difference cannot be calculated. 3. Each building needs both its before-upgrade and after-upgrade annual water use, linked by building.

Answer

The file lacks a baseline for each building. It must contain matched before- and after-upgrade water-use values for every observed building.
Learning from data 39
A forestry study planted \(600\) seedlings and, three years later, measured the heights of the \(418\) seedlings still alive. A report claims that the height data describe all \(600\) originally planted seedlings after three years. Explain the flaw in that statement.

Hints

- Compare the original group with the group that could actually be measured later. - Identify which observational units have no response value. - Limit the conclusion to the population represented by the available data.

Solution

1. Height after three years was recorded only for seedlings that survived. 2. The \(182\) seedlings that died have no three-year height measurements and may differ systematically from the survivors. 3. The observed height distribution describes the surviving seedlings, not all originally planted seedlings.

Answer

The data describe only the \(418\) surviving seedlings. They cannot represent the three-year heights of all \(600\) originally planted seedlings because the nonsurvivors are missing.
Learning from data 40
A file contains the daily high temperatures for \(30\) consecutive days, but the rows were randomly reordered and the dates were removed. A researcher wants to find the longest run of consecutive days with a high above \(90\,{}^\circ\text{F}\). Explain why the values alone are insufficient and identify the information that must be restored.

Hints

- Decide whether the requested result depends only on the set of values or also on their order. - Consider what random reordering preserves and what it destroys. - Name the metadata needed to reconstruct adjacency.

Solution

1. A run of consecutive days depends on the chronological order of the observations. 2. Randomly reordered temperatures preserve the one-variable distribution but destroy the adjacency relationships between days. 3. The date or original sequence position for every temperature must be restored before the longest run can be determined.

Answer

The temperature values preserve the distribution but not which days were consecutive. Restore each observation’s date or chronological position before finding the longest run.
Learning from data 41
A city calculates the share of trash it recycled each month from two files: one with recycled amounts and one with total trash. The records are below. <table><thead><tr><th>Month</th><th>Recycled (tons)</th><th>All trash (tons)</th></tr></thead><tbody><tr><td>June</td><td>120</td><td>600</td></tr><tr><td>July</td><td>150</td><td>750</td></tr></tbody></table> A report calculates June's rate using \(120\) tons recycled and \(750\) tons of total trash. What went wrong? Which two amounts should be paired for June, and what information should the city use to match rows from the two files?

Hints

- Locate the two amounts used in the report. Where does each appear in the table? - What should the two measurements have in common to form one month's rate? - If the files sort their rows differently, what information would still let you connect the right records?

Solution

1. The report paired June's recycled amount with July's total, so the two values refer to different months. 2. June's matching values are \(120\) tons recycled and \(600\) tons of total trash. 3. The city should match records using the month, not their row positions in the separate files.

Answer

The report combined different months. Pair June's \(120\) tons recycled with its \(600\) tons of total trash; match the two files by month.
Learning from data 42
An air-quality monitor records one value every minute from \(6\) a.m. to \(10\) p.m. but only one value per hour overnight. An analyst makes an unweighted distribution of all recorded values and calls it “the distribution of air quality over a full day.” Explain the weighting problem and give one way to make equal amounts of clock time contribute equally.

Hints

- Compare how many rows are contributed by daytime and overnight hours. - Decide whether the intended observational units are readings or equal units of time. - Consider a data structure in which every hour contributes the same amount.

Solution

1. Daytime hours contribute many more recorded values than overnight hours, so the unweighted distribution describes recorded readings rather than equal portions of time. 2. Daytime conditions are overrepresented because of the higher sampling frequency. 3. One solution is to summarize the data into equal-length time intervals, such as one value per hour, before forming the distribution. Equivalent time-based weights could also be used.

Answer

The unweighted data overrepresent daytime because readings are taken more frequently then. Use one summary per equal time interval, or apply time-based weights, so each hour contributes equally.

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