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Chapter 7: Problem 56

How might stratification be used in each of the following sampling problems? a. A survey of household expenditures in a city. b. A survey to examine the lead concentration in the soil in a large plot of land. c. A survey to estimate the number of people who use elevators in a large building with a single bank of elevators. d. A survey of programs on a television station, taken to estimate the proportion of time taken up by advertising on Monday through Friday from 6 P.M. until 10 P.M. Assume that 52 weeks of recorded broadcasts are available for analysis.

Short Answer

Expert verified
Use stratification based on relevant subgroups: income or regions for expenditures, land sections for soil, time/floor for elevator use, and days/times for TV programs.

Step by step solution

01

Understanding Stratification

Stratification in sampling involves dividing a population into distinct subgroups (strata) that share similar characteristics, allowing for more precise and representative samples.
02

Stratification for Household Expenditure Survey

For household expenditures in a city, stratify by income levels, geographic regions, or household size. By dividing the population based on these characteristics, you can ensure diverse representation and improve the reliability of your expenditure data.
03

Stratification for Lead Concentration in Soil

Divide the large plot of land into sections based on landscape features, elevation, or usage areas. For example, separate areas near roads, residential areas, and undisturbed sections to accurately assess lead distribution.
04

Stratification for Elevator Usage Survey

Stratify the survey based on the floor where people are located or the time of day. You might consider separating data from peak hours and non-peak hours to account for different usage patterns.
05

Stratification for Television Program Survey

Divide the survey based on each weekday and segmented time slots or program types. By stratifying by time and content, you accurately evaluate the proportion of advertising during the specified hours.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Household Expenditures Survey
When conducting a Household Expenditures Survey, the goal is to gather data that accurately represents spending patterns within a city. Stratified sampling can greatly enhance the reliability of such data by ensuring diverse representation. For instance, you could divide households into strata based on key characteristics such as:
  • Income levels, which captures different spending abilities.
  • Geographic regions within the city to reflect possible cost of living variations.
  • Household size, as family size often affects expenditure patterns.
This stratified approach ensures that every significant subgroup is represented, allowing for precise comparisons and reliable insights into household spending behavior across various demographics.
Lead Concentration Survey
In conducting a Lead Concentration Survey on a large plot of land, stratification becomes crucial to account for varying levels of lead across different areas. You can stratify the land based on landscape characteristics and potential pollution sources:
  • Elevation levels, as lead concentration might differ with changes in altitude.
  • Proximity to roads, because areas adjacent to busy roads might have higher lead accumulations.
  • Usage areas, like residential versus undisturbed land, where human activity may impact lead levels.
This method enables a more accurate assessment of lead concentration, letting researchers pinpoint specific areas with potentially harmful levels.
Elevator Usage Survey
An Elevator Usage Survey aims to determine the number of people using elevators in a large building, which can be tricky due to fluctuating usage patterns. Stratified sampling helps in understanding these variations. Consider stratifying the survey data by:
  • Floor levels, to account for varied elevator demand depending on the height or specific business on each floor.
  • Time of day, distinguishing between peak times (like morning rush) and off-peak hours when usage patterns differ.
This stratification offers a detailed view of elevator use, which can be invaluable for planning maintenance schedules or improving system efficiency.
Television Program Survey
Evaluating the proportion of time taken up by advertising for a Television Program Survey requires a methodical approach. Stratified sampling allows for a more detailed analysis by grouping data based on:
  • Different weekdays, because program types and advertisement slots vary each day.
  • Time slots, like early evening versus late night, as advertisement frequency may change.
  • Program genres, since different types of shows might have varied advertisement volumes.
By examining these strata, researchers can gain a nuanced understanding of advertisement distribution, allowing them to identify trends specific to certain times and program types.

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Most popular questions from this chapter

This problem introduces the concept of a one-sided confidence interval. Using the central limit theorem, how should the constant \(k\) be chosen so that the interval \(\left(-\infty, \bar{X}+k s_{\bar{X}}\right)\) is a \(90 \%\) confidence interval for \(\mu-\) i.e., so that \(P(\mu \leq \bar{X}+\) \(\left.k s_{\bar{X}}\right)=.9 ?\) This is called a one-sided confidence interval. How should \(k\) be chosen so that \(\left(\bar{X}-k s_{\bar{X}}, \infty\right)\) is \(95 \%\) one-sided confidence interval?

The designer of a sample survey stratifies a population into two strata, H and L. H contains 100,000 people, and L contains \(500,000 .\) He decides to allocate 100 samples to stratum \(\mathrm{H}\) and 200 to stratum \(\mathrm{L},\) taking a simple random sample in each stratum. a. How should the designer estimate the population mean? b. Suppose that the population standard deviation in stratum \(\mathrm{H}\) is 20 and the standard deviation in stratum L is 10. What will be the standard error of his estimate? c. Would it be better to allocate 200 samples to stratum \(\mathrm{H}\) and 100 to stratum \(\mathrm{L} ?\) d. Would it be better to use proportional allocation?

For a random sample of size \(n\) from a population of size \(N,\) consider the following as an estimate of \(\mu:\) $$\bar{X}_{c}=\sum_{i=1}^{n} c_{i} X_{i}$$ where the \(c_{i}\) are fixed numbers and \(X_{1}, \ldots, X_{n}\) is the sample. a. Find a condition on the \(c_{i}\) such that the estimate is unbiased. b. Show that the choice of \(c_{i}\) that minimizes the variances of the estimate subject to this condition is \(c_{i}=1 / n,\) where \(i=1, \ldots, n\)

This problem presents an algorithm for drawing a simple random sample from a population in a sequential manner. The members of the population are considered for inclusion in the sample one at a time in some prespecified order (for example, the order in which they are listed). The \(i\) th member of the population is included in the sample with probability \(\frac{n-n_{i}}{N-i+1}\) "where \(n_{i}\) is the number of population members already in the sample before the ith member is examined. Show that the sample selected in this way is in fact a simple random sample; that is, show that every possible sample occurs with probability $$\frac{1}{\left(\begin{array}{l}N \\\n\end{array}\right)}$$

How would you respond to a friend who asks you, "How can we say that the sample mean is a random variable when it is just a number, like the population mean? For example, in Example A of Section 7.3.2, a simple random sample of size 50 produced \(\bar{x}=938.5 ;\) how can the number 938.5 be a random variable?鈥�
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