Examkin

Question

Difficulty: HardStatistical Sampling and Studies

A researcher wants to estimate the total number of households in a city of 40,00040,000 households that own at least one hybrid vehicle. The researcher randomly selects a sample of households to contact for a survey. The survey has a response rate of 40%40\%. Of the households that responded, 15%15\% reported owning at least one hybrid vehicle. A follow-up study of a random sample of the non-respondents indicates that the proportion of non-respondents who own at least one hybrid vehicle is half the proportion of the respondents who do. Based on the results of the survey and the follow-up study, what is the estimated total number of households in the city that own at least one hybrid vehicle?

Answer: 4200 households

Answer

The estimated total number of households in the city that own at least one hybrid vehicle is 4,200.
The correct answer of 4,200 is found by dividing the population of 40,000 households into two strata: respondents (40%, or 16,000 households) and non-respondents (60%, or 24,000 households). The number of hybrid-owning households among respondents is estimated as 15% of 16,000, which is 2,400. The proportion of hybrid owners among non-respondents is half of the respondents' rate, or 7.5%. The number of hybrid-owning households among non-respondents is estimated as 7.5% of 24,000, which is 1,800. Summing these estimates yields 2,400 + 1,800 = 4,200.

Step-by-Step Solution

1
Calculate the size of the respondent and non-respondent strata in the population.
Respondent stratum: 16,00016,000 households; Non-respondent stratum: 24,00024,000 households.
Since the survey's response rate is 40%40\%, we estimate that 40%40\% of the 40,00040,000 households are potential respondents, and the remaining 60%60\% (24,00024,000 households) are potential non-respondents.
2
Determine the proportion of hybrid vehicle owners in each stratum.
Respondent proportion: 15%15\% (or 0.150.15); Non-respondent proportion: 7.5%7.5\% (or 0.0750.075).
The respondent proportion is given as 15%15\%. The follow-up study indicates the non-respondent proportion is half of the respondent proportion, which is 12×15%=7.5%\frac{1}{2} \times 15\% = 7.5\%.
3
Calculate the estimated number of households owning a hybrid vehicle in each stratum and sum them.
2,400+1,800=4,2002,400 + 1,800 = 4,200 households.
Multiply each stratum size by its respective proportion: 16,000×0.15=2,40016,000 \times 0.15 = 2,400 and 24,000×0.075=1,80024,000 \times 0.075 = 1,800. Adding these gives the total estimate of 4,2004,200.

Key Concept

Generalizing from stratified random samples to estimate population parameters while accounting for non-response bias.
Rate this question

Topics

Statistical Sampling and Studies

More questions from this topic