Question

3.62 Employer-Based Health Insurance A report from a Gallup poll \(^{29}\) in 2011 started by saying, "Forty-five percent of American adults reported getting their health insurance from an employer...." Later in the article we find information on the sampling method, "a random sample of 147,291 adults, aged 18 and over, living in the US," and a sentence about the accuracy of the results, "the maximum margin of sampling error is ±1 percentage point." (a) What is the population? What is the sample? What is the population parameter of interest? What is the relevant statistic? (b) Use the margin of error \(^{30}\) to give an interval showing plausible values for the parameter of interest. Interpret it in terms of getting health insurance from an employer.

Short answer

The population is all American adults, the sample is the 147,291 adults included in the Gallup poll. The population parameter of interest is the actual percentage of American adults getting their health insurance from an employer while the relevant statistic is the 45% reported from the survey. The plausible values for the parameter of interest, due to the margin of error, range from 44% to 46%, suggesting that the actual percentage of American adults who get their health insurance from an employer falls within this range.

Step-by-step solution

Identification of elements

(a) The population here refers to all American adults. The sample refers to the 147,291 adults chosen for the Gallup poll. The population parameter of interest is the actual percentage of American adults getting their health insurance from an employer. The relevant statistic is the 45% reported result from the survey.

Calculation of margin of error

(b) The margin of error is ±1%. This means the true population proportion is likely to fall between the range of (the percentage reported in the survey - margin of error) and (the percentage reported in the survey + margin of error). So in numbers, it's between (45 - 1)% = 44% and (45 + 1)% = 46%.

Interpretation of results

Therefore, we can interpret this by saying that there's a good chance that the actual percentage of all U.S. adults who get their health insurance from an employer is between 44% and 46%.

Key concepts

Gallup Poll Methodology

The Gallup poll is a well-recognized methodology for survey research that aims to represent the views or characteristics of a broad population by sampling a subset of that population. The process typically involves selecting respondents randomly to minimize bias and ensure that every individual in the population has an equal chance of being chosen. This random sample then provides data that can be extrapolated to the whole population within a known margin of error. The size of the sample, like the 147,291 adults in the health insurance poll, affects the confidence in the results and the margin of error.

In the context of the Gallup poll mentioned in the exercise, researchers gathered data on American adults' health insurance coverage. Random sampling aids in producing results that are reflective of the population as a whole, provided that the sample is sufficiently large and randomly selected. Moreover, these methodologies often include steps for reducing non-sampling errors, such as data processing mistakes or response errors, thereby enhancing the accuracy of the poll findings.

Population and Sample in Statistics

The terms 'population' and 'sample' are fundamental to understanding statistical analysis. The 'population' refers to the entire group about which information is wanted. In our exercise, the population is all American adults. On the other hand, a 'sample' is a subset of the population that is actually observed and measured — for the Gallup poll, this was the 147,291 adults surveyed.

The primary reason for sampling is that it usually is impractical or impossible to study the whole population. By carefully selecting a representative sample, statisticians can infer characteristics about the entire population. The challenge is to ensure that the sample accurately reflects the broader population it aims to represent, which is where random sampling and proper survey design play crucial roles.

Population Parameter

A 'population parameter' is a value that represents a characteristic of the entire population, such as a mean or proportion. It is usually unknown and is what researchers attempt to estimate by using statistics derived from the sample. In the case of the Gallup poll, the population parameter of interest is the true percentage of American adults who get their health insurance from an employer.

The statistic that corresponds to this parameter in our example is the reported 45% from the Gallup survey, which comes from the sample data. Analysts use this sample statistic to estimate the population parameter. However, it is important to note that because the sample only includes a subset of the population, sampling error will inevitably result in some level of uncertainty around the population parameter estimate.

Margin of Error Calculation

The 'margin of error' reflects the range within which the true population parameter is expected to fall with a certain level of confidence. It is calculated based on the sample size, the variance within the sample, and the level of confidence desired for the interval. In the exercise provided, the margin of error is given as ±1 percentage point.

To calculate an interval for plausible values of the population parameter, one would take the sample statistic (say, a proportion like the 45% from the poll) and create a range by subtracting and adding the margin of error. This gives us an interval estimate, which, in this case, is 44% to 46%. This suggests that if the poll were to be conducted on the entire population, the true percentage of American adults getting insurance from an employer would likely fall within that range. This calculation is essential in conveying the uncertainty of the sample statistic as an estimate of the population parameter.