Question

Exercise 8.17 discussed a research poll conducted for Fox News by Opinion Dynamics concerning opinions about the number of legal immigrants entering the United States. \(^{3}\) Suppose you were designing a poll of this type. a. Explain how you would select your sample. What problems might you encounter in this process? b. If you wanted to estimate the percentage of the population who agree with a particular statement in your survey questionnaire correct to within \(1 \%\) with probability .95, approximately how many people would have to be polled?

Short answer

Question: Discuss two important aspects of conducting a research poll on opinions about the number of legal immigrants entering the United States, including sample selection and the required sample size. Answer: A crucial aspect of conducting such a research poll is sample selection. In this case, we would use a simple random sampling technique to ensure that each individual in the population has an equal probability of being selected for the sample. This method involves defining the target population, creating a sampling frame, and using a random number generator to select the participants. Some potential problems in the sample selection process include creating a complete sampling frame, non-response bias, selection bias, language barriers, and time and cost constraints. Another important aspect is determining the required sample size. To achieve a desired level of accuracy (1%) with a confidence level of 0.95 (95%), we can use a sample size formula. Assuming a worst-case scenario with a proportion (P) of 0.5, the estimated required sample size would be approximately 9604 people to estimate the percentage of the population agreeing with a particular statement in the survey.

Step-by-step solution

Explanation of sample selection method

To select a sample for this poll, we should choose a sampling method that is both statistically valid and provides an accurate representation of the population. A common and efficient method for this purpose is the simple random sampling technique. This method ensures that each individual or unit in the population has an equal probability of being selected in the sample. To perform simple random sampling, we can follow these steps: 1. Define the target population, which in this case are the individuals in the United States with opinions about the number of legal immigrants entering the country. 2. Create a complete list or a sampling frame of all the individuals in the population. This can be done using a database, public records, or any other source containing information about the target population. 3. Assign a unique identification number to each individual in the sampling frame. 4. Use a random number generator to obtain a set of random identification numbers. 5. Select the individuals corresponding to these random numbers for the sample.

Potential problems in the sample selection process

There may be several problems encountered in this process, including: 1. Creating a complete and accurate sampling frame: It may be difficult to obtain information about all the individuals in the population, as some individuals may not appear in public records, databases, or other sources. 2. Non-response bias: Some individuals selected for the sample may be unavailable or unwilling to participate in the survey, leading to non-response bias. This problem may be more pronounced in internet or telephone surveys. 3. Selection bias: If certain segments of the population have a higher probability of being selected in the sample, it may lead to biased results. This problem can occur if the sampling frame excludes or under-represents particular groups. 4. Language barriers: The participants in the survey may not be proficient in English or the language of the survey, which may limit their ability to accurately express their opinions. 5. Time and cost constraints: Obtaining a large and representative sample can be time-consuming and expensive. This may restrict the ability to carry out the survey in an effective manner.

Required sample size calculation

In order to estimate the required sample size for achieving a desired level of accuracy (1%) with a confidence level of 0.95 (95%), we can use the following formula for the sample size: $$n = \frac{Z^2 \times P \times (1-P)}{E^2}$$ Where, n = sample size Z = the z-score corresponding to the desired confidence level (1.96 for 95% confidence) P = the proportion or percentage of the population who agree with a particular statement (unknown) E = the desired level of accuracy (0.01 or 1%) Since we do not know the true proportion P, we can assume the worst-case scenario, which is P = 0.5 (50% agree and 50% disagree). This value of P will lead to the largest possible sample size to be on the safe side: $$n = \frac{(1.96)^2 \times 0.5 \times (1-0.5)}{(0.01)^2}$$ Calculating the sample size: $$n \approx 9604$$ In this case, about 9604 people would have to be polled to estimate the percentage of the population agreeing with a particular statement in the survey with a 1% margin of error and a 95% confidence level.

Key concepts

Simple Random Sampling

Simple random sampling is a fundamental concept in statistical sampling that ensures every individual in the population has an equal chance of being included in the sample. This method is crucial for obtaining a representative sample that accurately reflects the overall population. Imagine every eligible voter in a small town has their name put into a hat, and we randomly pull out a number of names to decide who will respond to our survey on local issues. In a more technical example, suppose we have a list of all high school students in a district, and we use a random number generator to pick which students will participate in a study about their study habits.

Applying simple random sampling effectively can help eliminate selection bias, where certain segments of the population might inadvertently be sampled more frequently, skewing the results. For instance, if we only surveyed people arriving at a mall, we might miss those who prefer to shop online, impacting the results of a survey aimed at understanding shopping preferences.

Keep in mind that even with random selection, other types of errors can occur, such as non-response bias, which is discussed in a following section.

Non-response Bias

Non-response bias is a major concern in survey research and occurs when the individuals selected for the sample don’t respond at the same rate. This bias can profoundly affect survey results, introducing systematic differences that can misrepresent the population's true characteristics. Think of a situation where a survey is sent by mail, and those who feel strongly about the topic are more likely to return it. Their responses might be overrepresented, making it appear as though their opinion is more common within the general population than it actually is.

To minimize non-response bias, researchers can use strategies to encourage participation, such as follow-up reminders, incentives, and ensuring the survey is accessible and brief. Consideration of the communication channel is crucial; for example, younger demographics might be more responsive to an online survey than a traditional mail-in questionnaire. Despite these efforts, researchers need to acknowledge this potential bias during the analysis and interpretation of study findings.

Statistical Sampling

Statistical sampling is a broad term that encompasses various techniques used to select a part of the population to make inferences about the whole. It includes methods such as simple random sampling, stratified random sampling, systematic sampling, and cluster sampling, each with specific applications and benefits. For example, stratified random sampling might be used when the population has distinct subgroups that all need to be represented properly, like dividing a nation's population by states before randomly sampling within each state.

While these methods aim to produce a smaller group that represents the larger population with high accuracy, challenges like creating a comprehensive sampling frame, dealing with missing data, and avoiding bias must be handled carefully. Employing statistical sampling is a balance of statistical theory, practical logistics, and cost considerations. Notably, the effectiveness of the sampling method chosen can significantly impact the generalizability and credibility of the research conclusions.

Confidence Level and Margin of Error

The concepts of confidence level and margin of error are inseparable in the context of survey research and indicative of the precision of the survey results. The confidence level reveals the degree of certainty that the survey results represent the true population parameter. For instance, a 95% confidence level means that if we were to take 100 different samples and conduct the survey, approximately 95 of those samples would produce results within a particular range around the population parameter.

The margin of error is the range on either side of a survey result that illustrates where the true value for the population may lie. A smaller margin of error indicates a more precise estimate but typically requires a larger sample size, hence greater resource expenditure. For example, if a survey result states that 60% of people like a new product with a 3% margin of error at a 95% confidence level, it suggests that we can be pretty sure (95 out of 100 times) that the true percentage lies between 57% and 63% of all people.