Question

Occasionally, when I fill my car with gas, I figure out how many miles per gallon my car got. I wrote down those results after 6 fill-ups in the past few months. Overall, it appears my car gets \(28.8\) miles per gallon. a) What statistic have I calculated? b) What is the parameter I'm trying to estimate? c) How might my results be biased? d) When the Environmental Protection Agency (EPA) checks a car like mine to predict its fuel economy, what parameter is it trying to estimate?

Short answer

a) Sample mean. b) True average mpg for your car. c) Results may not reflect usual driving conditions. d) True average mpg for the car model in standard conditions.

Step-by-step solution

Understanding the Statistic

The statistic is a numerical value derived from data collected from a sample. In this case, calculating the average miles per gallon (mpg) from 6 fill-ups provides a sample mean. Therefore, the statistic here is the sample mean of 28.8 mpg.

Identifying the Parameter of Interest

The parameter is a numerical value that summarizes a characteristic for an entire population. In this context, the parameter you are trying to estimate with the sample data (6 fill-ups) is the true average miles per gallon (mpg) your car gets under all possible driving conditions and fill-ups.

Considering Potential Bias

Bias can occur if the sample is not representative of the population or if there are systematical errors in measurement. Here, your results might be biased if, for example, you drive differently during fill-ups than usual driving conditions, or measure mpg inconsistently, leading to results that may not reflect the true average mpg.

Parameter Estimated by EPA

The EPA's goal is to estimate a general parameter that represents the fuel economy of a particular car model across various driving conditions. This typically refers to the average miles per gallon (mpg) a car can achieve when driven in standard conditions that the EPA specifies.

Key concepts

Sample Mean

The sample mean is an essential concept in statistics, especially for understanding the average of a set of data collected from a sample. In this scenario, the sample mean is calculated by taking mile-per-gallon (mpg) data from 6 different fill-ups and finding their average. This value represents the typical mpg you experienced during those specific instances.
This concept is crucial because it provides a snapshot of what might be happening in the larger population, using data collected from a few examples. By computing the sample mean of 28.8 mpg, you've gathered a statistic that offers insight into your car's fuel efficiency during those recorded periods.
Overall, remember that the sample mean is limited to the data collected and may not perfectly reflect every possible scenario of driving conditions.

Parameter Estimation

Parameter estimation is a key part of statistics that aims to infer or predict the true population parameter based on sample data. In this context, the population parameter you seek to understand is the actual average mpg for your car under all possible driving conditions. This target is often unknown, so estimations are necessary.
Using the sample mean, we try to get close to this unknown value. However, it's important to realize that the estimate you get is dependent on many factors, including how you collect your data and any variations in your driving.
Effective parameter estimation helps you understand what to expect in broader circumstances if we could observe every instance. It's a method to bridge what you've seen (sample data) with what you want to understand (population characteristic).

Bias in Data

Bias in data is a pervasive issue that can skew results and lead to inaccurate conclusions. In your car's mpg calculation, bias might occur if the method of data collection does not truly represent overall driving conditions. For instance, if you fill up more often during long highway trips, your calculated mpg might be higher than usual.
Types of bias that could affect your results include:

• Selection Bias: You might be recording mpg in specific situations that don't cover the full spectrum of driving conditions.
• Measurement Bias: Inconsistent ways of measuring or calculating mpg could lead to skewed results.

To minimize bias, ensure that your sample represents typical conditions and that your measurement methods are consistent.

EPA Fuel Economy Testing

The EPA sets out to estimate a car's fuel economy parameter by testing vehicles in standardized conditions. When they evaluate fuel economy, they aim to measure the average mpg that a car model would achieve based on their specific testing protocols.
This parameter represents a generalized measure of fuel efficiency across various driving scenarios, enabling consumers to compare vehicles reliably. Their tests usually occur under controlled environments which might not mirror real-world conditions perfectly, but their goal is to offer a baseline for information on vehicle efficiency.
Keep in mind, however, that individual experiences may vary based on driving habits, maintenance, and environmental conditions. Nevertheless, the EPA's efforts provide crucial, standardized metrics for estimating fuel efficiency among different vehicles.