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Chapter 1: Problem 6

The California State University (CSU) system consists of 23 campuses, from San Diego State in the south to Humboldt State near the Oregon border. A CSU administrator wishes to make an inference about the average distance between the hometowns of students and their campuses. Describe and discuss several different sampling methods that might be employed. Would this be an enumerative or an analytic study? Explain your reasoning.

Short Answer

Expert verified
Use various sampling methods, and it's an analytic study.

Step by step solution

01

Understanding the Context

The problem involves determining the average distance between students' hometowns and their campuses in the CSU system, which consists of 23 campuses. This requires collecting a sample from the population of all students attending CSU.
02

Simple Random Sampling

In Simple Random Sampling, each student from the CSU system has an equal chance of being selected. This method ensures fairness and can be accomplished by assigning a number to each student and using a random number generator to select the sample.
03

Stratified Sampling

Stratified Sampling involves dividing the student population into different 'strata,' such as by campus or major, and then randomly sampling from each stratum. This ensures all segments of the student population are proportionately represented in the sample.
04

Cluster Sampling

In Cluster Sampling, the researcher might divide the CSU population into clusters based on campuses, then randomly select a few campuses, and measure the distances for all students in those campuses. This method can reduce costs and time by focusing on selected clusters.
05

Systematic Sampling

Systematic Sampling involves selecting every kth student from a lined-up list of all CSU students. For instance, if the list is 10,000 students long and we want 1,000 students, we'd choose every 10th student. This method is simple to implement if a well-organized list is available.
06

Enumerative vs Analytic Study

This situation represents an analytic study because the goal is to make inferences about the average distance beyond the specific sample, applying the results to the broader student population in future analyses.

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Key Concepts

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

Simple Random Sampling
Simple random sampling is like drawing names from a hat. Each student from the CSU system gets an equal chance of being selected. This method is highly valued for its fairness and simplicity. To implement it for the CSU campuses, you could assign a unique number to each student. Then, use a random number generator to pick your sample.
  • Ensures each student has an equal probability of being sampled.
  • Avoids bias, making it a pure representation of the population.
  • Requires a complete list of students, which can be large and complex to manage.
Simple random sampling is efficient, yet it requires a lot of groundwork, especially maintaining the up-to-date list of all students.
Stratified Sampling
Stratified sampling is powerful when the population has distinct layers. For CSU, this can mean dividing students based on campuses or majors. Each of these divisions is called a 'stratum'. Once you have your strata, you randomly sample students from each one. This ensures that every significant subgroup of the student population is represented in the sample.
  • Enhances precision by including diverse student backgrounds.
  • Useful if certain groups (like specific campuses) have unique traits.
  • Requires prior knowledge about student segments, such as demographics or fields of study.
Stratified sampling helps in better analyzing specific groups, leading to more insightful conclusions about the student body.
Cluster Sampling
Cluster sampling is an efficient alternative when managing a large population like CSU students. Here, every campus could be a cluster. Instead of sampling students from all campuses, randomly pick a few campuses and gather data from every student within those selected ones.
  • Cost-effective as it focuses only on selected clusters of campuses.
  • Reduces time and resources spent compared to sampling across all campuses.
  • May introduce bias if selected clusters have differences not represented in others.
Cluster sampling is great for logistical ease, though it needs careful cluster selection to maintain representation.
Systematic Sampling
Systematic sampling simplifies the sampling process. For CSU, imagine lining all students up and picking every kth person. Say, from 10,000 students, you'd pick every 10th to get a sample of 1,000. It's easier than simple random sampling, as you don't need a random number generator and merely require an ordered list.
  • Simplifies the sample selection process with minimal steps.
  • Ideal when you have an organized student list available.
  • Can introduce bias if there's a hidden pattern in the list order.
Systematic sampling strikes a balance between simplicity and efficient sampling, provided the list order doesn't unintentionally influence the sample.
Analytic Study
An analytic study, as in the CSU case, goes beyond merely understanding this specific sample. The goal is to draw conclusions that apply broadly to the entire student population, or even future ones. You aim to make inferences about the average distance between students' hometowns and campuses across CSU.
  • Focuses on presumptive conclusions applicable to broader populations.
  • Useful for projecting future trends based on current sample data.
  • Differs from enumerative studies that analyze the characteristics of the sample itself.
An analytic study is invaluable for strategic educational planning, giving insights that extend beyond immediate observations.

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Most popular questions from this chapter

The article "Determination of Most Representative Subdivision" (J. of Energy Engr., 1993: 43-55) gave data on various characteristics of subdivisions that could be used in deciding whether to provide electrical power using overhead lines or underground lines. Here are the values of the variable \(x=\) total length of streets within a subdivision: \(\begin{array}{rrrrrrr}1280 & 5320 & 4390 & 2100 & 1240 & 3060 & 4770 \\ 1050 & 360 & 3330 & 3380 & 340 & 1000 & 960 \\ 1320 & 530 & 3350 & 540 & 3870 & 1250 & 2400 \\ 960 & 1120 & 2120 & 450 & 2250 & 2320 & 2400 \\ 3150 & 5700 & 5220 & 500 & 1850 & 2460 & 5850 \\ 2700 & 2730 & 1670 & 100 & 5770 & 3150 & 1890 \\ 510 & 240 & 396 & 1419 & 2109 & & \end{array}\) a. Construct a stem-and-leaf display using the thousands digit as the stem and the hundreds digit as the leaf, and comment on the various features of the display. b. Construct a histogram using class boundaries 0,1000 , \(2000,3000,4000,5000\), and 6000 . What proportion of subdivisions have total length less than 2000 ? Between 2000 and 4000 ? How would you describe the shape of the histogram?

Observations on burst strength \(\left(\mathrm{lb} / \mathrm{in}^{2}\right)\) were obtained both for test nozzle closure welds and for production cannister nozzle welds ("Proper Procedures Are the Key to Welding Radioactive Waste Cannisters," Welding J., Aug. 1997: \(61-67)\) \(\begin{array}{lllllll}\text { Test } & 7200 & 6100 & 7300 & 7300 & 8000 & 7400 \\ & 7300 & 7300 & 8000 & 6700 & 8300 & \\ \text { Cannister } & 5250 & 5625 & 5900 & 5900 & 5700 & 6050 \\ & 5800 & 6000 & 5875 & 6100 & 5850 & 6600\end{array}\) Construct a comparative boxplot and comment on interesting features (the cited article did not include such a picture, but the authors commented that they had looked at one).

Temperature transducers of a certain type are shipped in batches of 50 . A sample of 60 batches was selected, and the number of transducers in each batch not conforming to design specifications was determined, resulting in the following data: \(\begin{array}{llllllllllllllllllll}2 & 1 & 2 & 4 & 0 & 1 & 3 & 2 & 0 & 5 & 3 & 3 & 1 & 3 & 2 & 4 & 7 & 0 & 2 & 3 \\ 0 & 4 & 2 & 1 & 3 & 1 & 1 & 3 & 4 & 1 & 2 & 3 & 2 & 2 & 8 & 4 & 5 & 1 & 3 & 1 \\ 5 & 0 & 2 & 3 & 2 & 1 & 0 & 6 & 4 & 2 & 1 & 6 & 0 & 3 & 3 & 3 & 6 & 1 & 2 & 3\end{array}\) a. Determine frequencies and relative frequencies for the observed values of \(x=\) number of nonconforming transducers in a batch. b. What proportion of batches in the sample have at most five nonconforming transducers? What proportion have fewer than five? What proportion have at least five nonconforming units? c. Draw a histogram of the data using relative frequency on the vertical scale, and comment on its features.

Every score in the following batch of exam scores is in the \(60 \mathrm{~s}, 70 \mathrm{~s}, 80 \mathrm{~s}\), or \(90 \mathrm{~s}\). A stem-and-leaf display with only the four stems \(6,7,8\), and 9 would not give a very detailed description of the distribution of scores. In such situations, it is desirable to use repeated stems. Here we could repeat the stem 6 twice, using \(6 \mathrm{~L}\) for scores in the low 60 s (leaves \(0,1,2,3\), and 4 ) and \(6 \mathrm{H}\) for scores in the high 60 s (leaves \(5,6,7,8\), and 9 ). Similarly, the other stems can be repeated twice to obtain a display consisting of eight rows. Construct such a display for the given scores. What feature of the data is highlighted by this display? \(\begin{array}{lllllllllllll}74 & 89 & 80 & 93 & 64 & 67 & 72 & 70 & 66 & 85 & 89 & 81 & 81 \\ 71 & 74 & 82 & 85 & 63 & 72 & 81 & 81 & 95 & 84 & 81 & 80 & 70 \\ 69 & 66 & 60 & 83 & 85 & 98 & 84 & 68 & 90 & 82 & 69 & 72 & 87 \\ 88 & & & & & & & & & & & & \end{array}\)

The sample data \(x_{1}, x_{2}, \ldots, x_{n}\) sometimes represents a time series, where \(x_{t}=\) the observed value of a response variable \(x\) at time \(t\). Often the observed series shows a great deal of random variation, which makes it difficult to study longerterm behavior. In such situations, it is desirable to produce a smoothed version of the series. One technique for doing so involves exponential smoothing. The value of a smoothing constant \(\alpha\) is chosen \((0<\alpha<1)\). Then with \(\bar{x}_{t}=\) smoothed value at time \(t\), we set \(\bar{x}_{1}=x_{1}\), and for \(t=2,3, \ldots, n, \bar{x}_{t}=\alpha x_{t}+(1-\alpha) \bar{x}_{t-1} .\) c. Substitute \(\bar{x}_{r-1}=\alpha x_{r-1}+(1-\alpha) \bar{x}_{r-2}\) on the right-hand side of the expression for \(\bar{x}_{n}\) then substitute \(\bar{x}_{t-2}\) in terms of \(x_{1-2}\) and \(\bar{x}_{1-3}\), and so on. On how many of the values \(x_{r}, x_{t-1}, \ldots, x_{1}\) does \(\bar{x}_{1}\) depend? What happens to the coefficient on \(x_{t-k}\) as \(k\) increases? d. Refer to part (c). If \(t\) is large, how sensitive is \(\bar{x}_{t}\) to the initialization \(\bar{x}_{1}=x_{1}\) ? Explain. [Note: A relevant reference is the article "Simple Statistics for Interpreting Environmental Data," Water Pollution Control Fed. J., 1981: 167-175.] a. Consider the following time series in which \(x_{t}=\) temperature \(\left({ }^{\circ} \mathrm{F}\right)\) of effluent at a sewage treatment plant on day \(t: 47,54,53,50,46,46,47,50,51,50,46\), \(52,50,50\). Plot each \(x_{t}\) against \(t\) on a two-dimensional coordinate system (a time-series plot). Does there appear to be any pattern? b. Calculate the \(\bar{x}_{t}\) 's using \(\alpha=.1\). Repeat using \(\alpha=.5\). Which value of \(\alpha\) gives a smoother \(\bar{x}_{t}\) series?
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