Question

Is a Car a Necessity? A random sample of \(n=1483\) adults in the US were asked whether they consider a car a necessity or a luxury, \({ }^{31}\) and we find that a \(95 \%\) confidence interval for the proportion saying that it is a necessity is 0.83 to \(0.89 .\) Explain the meaning of this confidence interval in the appropriate context.

Short answer

The 95% confidence interval of 0.83 to 0.89 indicates that we can be 95% confident that the actual percentage of adults in the US who consider a car a necessity falls between 83% and 89% based on the sampled data.

Step-by-step solution

Understanding Confidence Interval

A confidence interval is an estimated range of values which is likely to include an unknown population parameter. Here, it is used to determine the proportion of people who think a car is a necessity.

Applying Confidence Interval

The 95% confidence interval given is 0.83 to 0.89. This implies that if repeated samples were taken and the 95% confidence interval was calculated for each sample, the actual population parameter would be within the interval estimates 95% of the time.

Interpreting the Confidence Interval

The confidence interval 0.83 to 0.89 indicates that we can be 95% confident that the true proportion of all US adults who consider a car a necessity lies between 83% and 89%.

Key concepts

Statistical Inference

Statistical inference is a cornerstone of data analysis, allowing us to draw conclusions about a population based on a sample of data. It's a way of 'making sense' of data by using various methods to test hypotheses and estimate population characteristics. In the context of the given exercise, statistical inference is used to derive a conclusion about the entire US adult population’s opinion on cars being a necessity, based on the sample of 1,483 individuals.

Understanding Through Confidence Intervals

One of the primary tools of statistical inference is the confidence interval, which gives us a range in which we expect the true population parameter to fall a certain percentage of the time. For instance, in our car necessity question, we interpret the 95% confidence interval of 0.83 to 0.89 as a statistically informed estimate, indicating that we are 95% certain the actual proportion in the full population falls within this range. The confidence level (95% in this case) reflects how sure we are that the intervals calculated from different random samples will contain the true parameter.

Population Parameter Estimation

Population parameter estimation is the process of using sample data to estimate the characteristics of a larger population. In a statistical sense, parameters are numerical characteristics that summarize data for an entire population, like the mean or proportion. As individuals, we cannot survey an entire population due to constraints like time and cost, but we can estimate parameters like the mean or proportion using a representative sample.

The Role of Sample Data

The 1,483 US adults represent the sample data from which we can estimate the population parameter — in this case, the proportion who view a car as a necessity rather than a luxury. Estimation comes in two types: point estimation and interval estimation. Point estimation provides a single value as an estimate of the population parameter, while interval estimation gives a range (like the confidence interval) where the parameter is likely to fall. The latter is more informative as it also communicates the estimate's precision.

Sample Proportion

The sample proportion is a statistic that estimates the proportion of the population that has a particular characteristic, based on a sample drawn from that population. It's calculated by dividing the number of individuals in the sample with the characteristic by the total sample size. In our exercise, the characteristic of interest is considering a car a necessity.

Relevance in the Real World

For the surveyed sample of US adults, the calculation would involve the number who answered that they see a car as a necessity divided by 1,483, the total sample size. This gives us a sample proportion, which we then use to infer about the population's perspective. Because it's based on a sample, the sample proportion is subject to sampling variability—thus the need for a confidence interval, which acknowledges this variability and provides a range that the true population proportion is likely to fall within.