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) Find the sampling proportion. (b) Suppose that the survey had a response rate of \(90 \%\). Find the size \(n\) of the sample.

Step by step solution

01

Calculate Sampling Interval

The sampling interval is given as 100. That means, one student is selected after every 100 students. So the sampling proportion can be calculated as 1 Student every 100 Students.

02

Calculate Total Sample Size

Considering this sampling proportion, the total number of students that would be sampled from the 15,000 undergraduates would be \(\frac{15000}{100}\), which equals 150 students.

03

Adjust for Response Rate

Given that the response rate of the survey was 90%, therefore the effective sample size that responded would be \(90\%\) of 150, which will be \(0.90 * 150 = 135\).

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Key Concepts

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

Sampling Proportion

Systematic sampling is a straightforward way to draw a sample from a population, in this case, the undergraduate students at Tasmania State University. The method involves selecting every nth item from a list. Here, the sampling proportion refers to the fraction of the total population that is selected as the sample.
In our specific example, one student is chosen for every 100 in the list. This means that the sampling proportion is calculated as:

• Sampling Proportion = \(\frac{1}{100}\)

This indicates that 1% of the entire undergraduate population is being sampled. Understanding this concept is essential as it defines how representative the sample is and ensures that each part of the population has an equal chance of being included.

Response Rate

The response rate of a survey indicates the percentage of sampled individuals who actually complete the survey. It is a vital measure of survey quality. In this example, a response rate of 90% shows that most of the students selected completed the survey.
A high response rate is crucial as it suggests:

• Greater reliability of the survey results
• Lower non-response bias, which can skew the data

To find the effective sample size, which accounts for those who did not respond, you calculate as follows:

• Effective Sample Size = Response Rate \(\times\) Total Sample Size
• Here, that's \(0.90 \times 150 = 135\)

Thus, the 135 students who responded represent the actual dataset used for analysis.

Sample Size

Sample size is the number of observations or replicates included in a statistical sample. In the case of our exercise, the total sample size before considering the response rate influences how robust your conclusions can be.
The sample size is determined by dividing the total population by the interval of selections used in systematic sampling:

• Sample Size = \(\frac{Total \, Population}{Sampling \, Interval}\)
• In this example, \(\frac{15000}{100} = 150\)

The larger the sample size, the greater the reliability of the study, as it usually provides a closer estimate of the population parameter. However, understanding sample size is not just about the total number; adjustments might be required based on response rates to cover non-respondents and ensure statistical validity.