91Ó°ÊÓ

Refer to the following story (see also Exercise 32): The Dean of Students at Tasmania State University wants to determine how many undergraduates at TSU are familiar with a new financial aid program offered by the university. There are 15,000 undergraduates at \(T S U,\) so it is too expensive to conduct a census. The following sampling method, known as systematic sampling, is used to choose a representative sample of undergraduates to poll. Start with the registrar's alphabetical listing containing the names of all undergraduates. Randomly pick a number between 1 and \(100,\) and count that far down the list. Take that name and every 100 th name after it. For example, if the random number chosen is \(73,\) then pick the \(73 \mathrm{rd}, 173 \mathrm{rd}, 273 \mathrm{rd},\) and so forth, names on the list. (a) Explain why the method used for choosing the sample is not simple random sampling. (b) If \(100 \%\) of those responding claimed that they were not familiar with the new financial aid program offered by the university, is this result more likely due to sampling variability or to sample bias? Explain.

Step by step solution

01

Understanding Systematic Sampling versus Simple Random Sampling

In systematic sampling, the sample is selected according to a specific rule or pattern, for instance, every 100th name on the list. In simple random sampling, each unit of the population has an equal chance of being selected, and selections are made independently. The key difference between these two methods is the randomness and independence of the selection process.

02

Explanation of why the Sampling Method is not Simple Random Sampling

With the method described in the exercise, not all undergraduates have an equal chance of being selected. The chance of being selected depends on the position of the name within every 100 names. Thus, the sampling method is not simple random sampling.

03

Understanding Sampling Variability and Sampling Bias

Sampling variability refers to the expectation that different samples from the same population will yield different results, simply due to chance. In contrast, sampling bias occurs when the sampling method used introduces a systemic bias into the sample, which can lead to misrepresentative results about the population.

04

Determine the Cause of the Result

If 100% of those responding claimed that they were not familiar with the new financial aid program offered by the university, it doesn't indicate sampling variability, as the result is not different from sample to sample. Instead, this could be an indication of sample bias if there's a systemic reason related to the sampling method that would make the selected individuals less likely to know the new financial aid program. However, without more information, it is also possible that the program is indeed not known among all undergraduates.

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Key Concepts

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

Sampling Bias

Sampling bias occurs when some members of a population are less likely to be included in the sample than others, which leads to unrepresentative data. This type of bias results in conclusions that do not accurately reflect the true nature of the population.
In the context of the exercise, if every 100th name is chosen, students whose names appear less frequently at these intervals have less chance of being included. If there is any systematic factor affecting the alphabetical list, like students with last names starting with late alphabet letters belonging to certain departments or year levels, they might end up being underrepresented.
Therefore, sampling bias can skew results, by excluding certain opinions or experiences, leading to potentially erroneous conclusions about the familiarity with the financial aid program.

Simple Random Sampling

Simple random sampling is a method where every member of the population has an equal and independent chance of being selected. This ensures that the sample represents the population fairly without any systemic exclusion or preference.
In the exercise, simple random sampling was not used because the selection depended on a systematic method (every 100th name), not offering equal opportunities for every member to be chosen. Unlike systematic sampling, simple random sampling would require a completely independent selection, such as drawing names from a hat or using a computer to randomly select students without following a sequence.
This method minimizes bias and increases the reliability of the data collected, as each person has an equal chance of being a part of the sample, capturing a more authentic snapshot of the population's opinions or knowledge levels.

Sampling Variability

Sampling variability refers to the natural differences that occur between different samples drawn from the same population. Even if a sampling method is properly designed, differences in sample outcomes are expected due to random chance.
In scenarios where there is variability, results across samples can vary: some might know about the aid program while others might not, just by probability.
However, in the exercise scenario where 100% of respondents were unfamiliar with the program, it suggests more than just normal variability, possibly indicating bias. Variability would generally show a range of responses assuming awareness varies among the student body. Nonetheless, consistent results across various samples can decrease questioning around sampling variability and increase the suspicion of other factors, like sampling bias.