Surveys and Sampling

Population: The entire group of individuals, items, or data points that you want to study. For example, all registered voters in a country, all students in a school, or all products manufactured by a company.

Sample: A subset of the population that is actually surveyed or measured. For example, 1,000 registered voters, 100 students, or 50 products.

It's important to understand that we rarely survey an entire population because:

• It's often impractical or impossible to survey everyone
• It's usually unnecessary - a well-designed sample can provide accurate results
• It's more cost-effective to survey a smaller group

There are several methods for selecting a sample from a population. The method used affects how well the sample represents the population.

• Random Sampling: Every member of the population has an equal chance of being selected. This is the ideal method for ensuring a representative sample.
• Stratified Sampling: The population is divided into subgroups (strata) based on certain characteristics, and then random samples are taken from each subgroup.
• Systematic Sampling: Every nth member of the population is selected (e.g., every 10th person on a list).
• Convenience Sampling: Members of the population are selected based on their availability or ease of access. This method often leads to biased results.
• Volunteer Sampling: Members of the population self-select to participate. This method often leads to biased results.

On the SAT, you'll need to evaluate whether a sampling method is appropriate for making inferences about a population.

A point estimate is a single value used to estimate a population parameter. For example:

• The sample mean is a point estimate of the population mean
• The sample proportion is a point estimate of the population proportion

Example: If a survey of 1,000 registered voters finds that 45% support a particular candidate, then 45% is a point estimate of the percentage of all registered voters who support that candidate.

The margin of error is a measure of the uncertainty in a point estimate. It represents the range of values within which the true population parameter is likely to fall.

On the SAT, you'll often see survey results presented with a margin of error, such as "45% ± 3%." This means that the true percentage in the population is likely to be between 42% and 48%.

The margin of error depends on several factors:

• Sample Size: Larger samples generally have smaller margins of error
• Population Size: For very large populations, the margin of error depends mainly on the sample size
• Confidence Level: Higher confidence levels result in larger margins of error

While you won't need to calculate the margin of error on the SAT, you should understand how to interpret it and how it affects the conclusions you can draw from survey results.

A confidence interval is a range of values that is likely to contain the true population parameter. It is calculated by adding and subtracting the margin of error from the point estimate.

Example: If a survey finds that 45% of respondents support a candidate with a margin of error of 3%, the 95% confidence interval is 42% to 48%. This means we can be 95% confident that the true percentage of all voters who support the candidate is between 42% and 48%.

On the SAT, you'll need to understand how to interpret confidence intervals and use them to make inferences about populations.

When a survey asks yes/no questions or questions with categorical responses, the results are often reported as proportions or percentages.

Example: A survey of 500 students finds that 60% prefer online learning to traditional classroom learning. We can estimate that approximately 60% of all students prefer online learning, with some margin of error.

To estimate the number of individuals in a population with a certain characteristic:

$\mathrm{Estimated\; number}=\mathrm{Population\; size}\times \mathrm{Sample\; proportion}$

Example: If 60% of 500 students prefer online learning, and there are 2,000 students in the school, we can estimate that approximately 1,200 students (2,000 × 0.6) prefer online learning.

When a survey collects numerical data, the results are often reported as means or averages.

Example: A survey of 100 households finds that the average monthly electricity bill is $120. We can estimate that the average monthly electricity bill for all households is approximately $120, with some margin of error.

To estimate the total value for a population:

$\mathrm{Estimated\; total}=\mathrm{Population\; size}\times \mathrm{Sample\; mean}$

Example: If the average monthly electricity bill for 100 households is $120, and there are 10,000 households in the city, we can estimate that the total monthly electricity bills for all households is approximately $1,200,000 (10,000 × $120).

Surveys often compare different groups within a population. On the SAT, you'll need to understand how to interpret these comparisons and determine if differences between groups are statistically significant.

Example: A survey finds that 65% of men and 55% of women support a particular policy. The difference is 10 percentage points. To determine if this difference is meaningful, we need to consider the margin of error for each group.

If the margins of error overlap (e.g., men: 65% ± 5%, women: 55% ± 5%), we cannot conclude with confidence that there is a real difference between the groups. If the margins of error do not overlap, we can be more confident that there is a real difference.

The sampling frame is the list of individuals or items from which the sample is drawn. It defines the population that the survey results can be generalized to.

Example: If a survey is conducted by calling landline telephone numbers, the sampling frame is people with landline telephones. The results cannot be generalized to people who only have cell phones.

On the SAT, you'll need to identify the sampling frame and determine if it matches the population you're interested in.

Several types of bias can affect the representativeness of a sample:

• Selection Bias: The sample is not representative of the population because of how it was selected.
• Nonresponse Bias: People who choose not to respond to a survey may differ from those who do respond.
• Volunteer Bias: People who volunteer to participate in a survey may differ from those who don't volunteer.
• Response Bias: People may not answer survey questions honestly or accurately.

On the SAT, you'll need to identify these biases and determine how they affect the generalizability of survey results.

When evaluating survey claims on the SAT, ask yourself these questions:

• Who was surveyed? (What is the sampling frame?)
• How were they selected? (What sampling method was used?)
• How many people responded? (What is the sample size?)
• What is the margin of error?
• To what population is the claim being generalized?
• Is the sampling frame representative of that population?

Example: A survey claims that "80% of Americans support a new policy." If the survey was conducted by calling landline telephones and only included registered voters, the claim should be qualified as "80% of registered voters with landline telephones support a new policy."

When survey results include margins of error, you need to consider the range of possible values.

Example: A survey finds that 45% of respondents support a candidate with a margin of error of 3%. This means the true percentage is likely to be between 42% and 48%.

If another survey finds that 48% of respondents support the same candidate with a margin of error of 3%, we cannot conclude that support has increased, because the confidence intervals overlap (42% to 48% and 45% to 51%).

To determine if a sample is representative of a population, compare the characteristics of the sample to the characteristics of the population.

Example: A survey of 100 students at a university finds that 60% are female. If 55% of all students at the university are female, the sample is reasonably representative in terms of gender.

However, if the survey was conducted only in the nursing department, where 80% of students are female, the sample would not be representative of the entire university.

Survey data can be used to make predictions about future events or behaviors.

Example: A survey finds that 70% of customers are satisfied with a product. If there are 1,000 customers, we can predict that approximately 700 customers will continue to use the product.

However, these predictions are not guaranteed. They are based on the assumption that the sample is representative of the population and that the behavior of the population will remain consistent over time.