Free-response practice · Question 1 style · 10 points

Carbon monoxide detector survey plans

The health department in Granite County wants to estimate the proportion of homes in the county that have a working carbon monoxide (CO) detector. The county has 52,000 homes in 260 neighborhoods, and it classifies every home as a single-family house, an apartment, or a mobile home. The department is considering three plans for collecting data. In Plans 1 and 2, an inspector visits each selected home and presses the test button on its CO detector to see whether the detector works.

Plan 1: Use a random number generator to select 4 of the 260 neighborhoods. Inspectors visit every home in the 4 selected neighborhoods.
Plan 2: Separate the county's list of homes by housing type. Select a simple random sample of 2% of the homes of each type, as shown in the table. Inspectors visit each selected home.
Plan 3: Mail a questionnaire to the 1,800 households that attended a free fire-safety workshop run by the county last year. The questionnaire asks whether the home has a working CO detector.

Homes in Granite County and the homes selected in Plan 2
Housing type	Homes in the county	Homes selected in Plan 2
Single-family house	34,000	680
Apartment	15,000	300
Mobile home	3,000	60
Total	52,000	1,040

Use the given information to respond to parts A, B, C, D, and E. Label any subparts (e.g., i and ii) that may be present.

A.

i. Identify the sampling method used in Plan 1.

ii. Identify the sampling method used in Plan 2.
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Score part A
- 1 point Identifies the sampling method in Plan 1 as a cluster random sample.Earns it: "cluster random sample"; "cluster sample, with neighborhoods as the clusters"Does not: "stratified random sample" (every home in a chosen neighborhood is visited, which makes the neighborhoods clusters, not strata); "simple random sample of neighborhoods" (the sample of homes is every home in the chosen neighborhoods, which is a cluster sample); "random sample" (does not name the method)
- 1 point Identifies the sampling method in Plan 2 as a stratified random sample, with housing types as the strata.Earns it: "stratified random sample"; "stratified sample by housing type"Does not: "cluster sample" (a separate random sample is taken within each housing type, so the types are strata); "simple random sample" (homes are not selected from the whole list at once); "systematic random sample" (homes are not chosen at a fixed interval from a random start)
Model response
i. Cluster random sample. The neighborhoods are the clusters, 4 are chosen at random, and every home in them is visited.

ii. Stratified random sample. The housing types are the strata, and a simple random sample is taken within each type.
B.

i. Explain one advantage of Plan 1 compared with Plan 2.

ii. Explain one advantage of Plan 2 compared with Plan 1.
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Score part B
- 1 point Explains a practical advantage of Plan 1: the homes are concentrated in only 4 neighborhoods, so inspectors spend much less time and money traveling than they would visiting homes spread across the county. A response may instead note that Plan 1 needs only a list of neighborhoods rather than a list of every home.Earns it: "Inspectors only travel to 4 neighborhoods instead of driving all over the county, so Plan 1 costs less."Does not: "Plan 1 is more random." (both plans use random selection); "Plan 1 gives a more accurate estimate." (no reason is given, and 4 neighborhoods may not represent the county well)
- 1 point Explains that Plan 2 guarantees each housing type is represented in proportion to the county. Because the proportion of homes with a working CO detector may differ by housing type, this gives a more precise or more representative estimate than 4 neighborhoods, which could happen to contain mostly one type of home. The response must refer to housing type or to how representative the sample is.Earns it: "Every housing type is sampled in proportion, so mobile homes cannot be left out the way they could be if the 4 neighborhoods happened to have none."; "Detector rates probably differ by housing type, so stratifying by type gives a more precise estimate."Does not: "Plan 2 is random and Plan 1 is not." (both plans are random); "Plan 2 has no bias." (vague; does not explain what Plan 2 does better)
Model response
i. Plan 1 is cheaper and faster to carry out. The homes visited are concentrated in 4 neighborhoods, so inspectors spend far less time driving than they would traveling to 1,040 homes scattered across the county.

ii. Plan 2 guarantees that each housing type is represented in the same proportion as in the county. The proportion of homes with a working detector probably differs by housing type, so stratifying gives a more precise estimate. In Plan 1, homes in the same neighborhood tend to be the same type, so 4 neighborhoods could easily include no mobile homes, or mostly apartments.
C.

i. Explain why the results of Plan 3 should not be generalized to all homes in Granite County.

ii. Determine whether Plan 3 is more likely to overestimate or underestimate the proportion of all homes in the county that have a working CO detector. Explain your reasoning.
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Score part C
- 1 point Explains that Plan 3 includes only households that attended a fire-safety workshop, so the homes were not randomly selected from all homes in the county and may differ from other homes. The results therefore do not represent all homes in Granite County.Earns it: "Only workshop attendees are surveyed. They were not randomly chosen from all homes and may not be like other households."Does not: "The sample is too small." (1,800 households is not small, and a larger sample would not fix how they were chosen); "People might lie on the questionnaire." (that is response bias, not a reason the sample cannot represent all homes)
- 1 point Determines that Plan 3 is more likely to overestimate the proportion and explains why: households that chose to attend a fire-safety workshop are likely to be more safety-conscious than typical households, so a higher proportion of them have working CO detectors. A determination without a reason does not earn the point.Earns it: "Overestimate, because people who go to a fire-safety workshop probably care more about safety and are more likely to have a working detector."Does not: "Overestimate." (no reason is given); "Underestimate, because some households will not mail the questionnaire back." (wrong direction, and nonresponse alone does not explain the direction)
Model response
i. Plan 3 surveys only households that chose to attend a fire-safety workshop. They were not randomly selected from all homes in the county, and they may differ from other households in ways related to CO detectors, so their results do not represent all homes in Granite County.

ii. Plan 3 is more likely to overestimate the proportion. People who attend a fire-safety workshop are probably more safety-conscious than typical residents, and they may have heard about CO detectors at the workshop, so a higher proportion of their homes would have working detectors.
D.

To save money, a staff member suggests replacing the inspector visits in Plan 2 with phone calls that ask, "Does your home have a working carbon monoxide detector?" Explain how this change could lead to response bias.
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Score part D
- 1 point Explains why phone answers would tend to differ from the truth in one direction: residents may say yes because they own a detector without knowing whether it still works, or because they do not want to admit they lack one, so the answers would overstate the proportion of homes with working detectors.Earns it: "People might say yes because they have a detector, even if its battery died long ago. The answers would overstate how many detectors work."Does not: "People might not answer the phone." (that is nonresponse, not response bias); "People could lie." (does not say why or in which direction the answers would be off)
Model response
Residents may answer yes because they own a CO detector, even if its battery is dead or they have never tested it, and some may say yes because they are embarrassed to admit they do not have one. The answers would tend to overstate the number of homes with a working detector, so the phone version would tend to overestimate the proportion.
E.

Determine a valid investigative question for the health department's study that could be answered with a confidence interval. Ensure all three components of a valid investigative question are included in your response.
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Score part E
- 1 point First component: identifies the variable of interest as whether a home has a working CO detector.Earns it: "whether a home has a working CO detector"; "homes that have a working carbon monoxide detector"Does not: "CO detectors" (does not say what is recorded about each home); "the number of CO detectors in a home" (the department records whether a home has a working detector, not how many detectors it has)
- 1 point Second component: identifies the parameter as the proportion of homes with a working CO detector and the goal of estimating it within a range of plausible values.Earns it: "an interval of plausible values for the proportion"; "between what two values does the proportion fall"Does not: "What proportion of homes have a working CO detector?" (names the parameter but not estimation within a range of values); "Is the proportion greater than 0.80?" (a question for a significance test, not an interval); "How many homes have detectors?" (a count, not a proportion)
- 1 point Third component: identifies the population as all homes in Granite County.Earns it: "all homes in Granite County"; "all 52,000 homes in the county"Does not: "all homes" (does not say which county); "everyone in Granite County" (the observational units are homes, not people); "the homes in the sample" (that is the sample, not the population)
Model response
What is an interval of plausible values for the proportion of all homes in Granite County that have a working carbon monoxide detector?

Your score: 0 of 10. Each point is earned on its own, the way the exam scores it.