Fuel Efficiency Analysis Of A Car Model Mileage Rating And Statistical Concepts
In today's world, fuel efficiency is a critical factor for car buyers. Car manufacturers often advertise the mileage ratings of their models to attract customers. However, it's essential to verify these claims through independent testing. This article delves into a statistical analysis of a car manufacturer's mileage rating for one of its models, exploring the concepts of average mileage, standard deviation, and sampling techniques.
A car manufacturer claims that its car model achieves an average mileage of 32 miles per gallon (mpg), with a standard deviation of $\sigma = 2 mpg$. To validate this claim, a consumer testing agency conducts a study. They select a random sample of 30 cars of this model and meticulously measure their fuel efficiency. This random sampling is crucial to ensure the results are representative of the entire population of cars of this model.
Understanding Key Statistical Concepts
Before diving into the analysis, let's define some key statistical concepts:
- Average (Mean): The average mileage represents the central tendency of the data. It is calculated by summing the mileage of each car in the sample and dividing by the total number of cars.
- Standard Deviation: The standard deviation measures the dispersion or spread of the data around the average. A smaller standard deviation indicates that the data points are clustered closely around the average, while a larger standard deviation suggests greater variability.
- Random Sample: A random sample is a subset of the population selected in such a way that each member of the population has an equal chance of being chosen. This ensures that the sample is representative of the population, minimizing bias in the results.
- Sample Size: The sample size refers to the number of observations in the sample. A larger sample size generally leads to more precise estimates of population parameters.
Now, let's explore how statistical methods can be used to analyze the mileage ratings obtained by the consumer testing agency. The agency's goal is to determine whether the sample data supports the manufacturer's claim of an average mileage of 32 mpg.
Hypothesis Testing
One common statistical technique used for this purpose is hypothesis testing. Hypothesis testing involves formulating two competing hypotheses:
- Null Hypothesis (H0): The null hypothesis assumes that there is no significant difference between the sample mean and the population mean. In this case, the null hypothesis would be that the average mileage of the car model is indeed 32 mpg.
- Alternative Hypothesis (H1): The alternative hypothesis contradicts the null hypothesis. It suggests that there is a significant difference between the sample mean and the population mean. In this case, the alternative hypothesis could be that the average mileage of the car model is either greater than or less than 32 mpg.
Test Statistic
To test the hypothesis, we need to calculate a test statistic. A test statistic is a numerical value that summarizes the sample data and provides evidence against the null hypothesis. The choice of test statistic depends on the type of data and the hypotheses being tested. In this scenario, since we are dealing with a sample mean and a known population standard deviation, we can use the z-statistic.
The z-statistic is calculated as follows:
z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))
Where:
- sample mean is the average mileage of the 30 cars in the sample.
- population mean is the manufacturer's claimed average mileage of 32 mpg.
- population standard deviation is the manufacturer's claimed standard deviation of 2 mpg.
- sample size is the number of cars in the sample, which is 30.
Significance Level and P-value
After calculating the test statistic, we need to determine the significance level and the p-value. The significance level (alpha) is the probability of rejecting the null hypothesis when it is actually true. A common significance level is 0.05, which means there is a 5% chance of making a Type I error (rejecting a true null hypothesis).
The p-value is the probability of obtaining a test statistic as extreme as or more extreme than the one observed, assuming the null hypothesis is true. A smaller p-value indicates stronger evidence against the null hypothesis.
Decision Rule
To make a decision about the hypotheses, we compare the p-value to the significance level. The decision rule is as follows:
- If the p-value is less than the significance level, we reject the null hypothesis.
- If the p-value is greater than the significance level, we fail to reject the null hypothesis.
Interpretation
If we reject the null hypothesis, it suggests that there is significant evidence that the average mileage of the car model is different from the manufacturer's claim of 32 mpg. If we fail to reject the null hypothesis, it suggests that there is not enough evidence to conclude that the average mileage is different from 32 mpg.
Let's assume the consumer testing agency finds that the average mileage of the 30 cars in the sample is 31 mpg. Using the formula for the z-statistic, we get:
z = (31 - 32) / (2 / sqrt(30))
z = -1 / (2 / 5.477)
z = -1 / 0.365
z = -2.74
The calculated z-statistic is -2.74. Now, we need to find the p-value associated with this z-statistic. Using a z-table or a statistical software, we find that the p-value for a two-tailed test (since we are interested in whether the average mileage is different from 32 mpg in either direction) is approximately 0.006.
If we use a significance level of 0.05, we see that the p-value (0.006) is less than the significance level (0.05). Therefore, we reject the null hypothesis. This suggests that there is significant evidence that the average mileage of the car model is different from the manufacturer's claim of 32 mpg.
It's important to note that fuel efficiency can be affected by various factors, including:
- Driving Habits: Aggressive driving, such as rapid acceleration and hard braking, can significantly reduce fuel efficiency.
- Road Conditions: Driving on hilly terrain or in stop-and-go traffic can also decrease mileage.
- Vehicle Maintenance: Regular maintenance, such as tire inflation and oil changes, can help optimize fuel efficiency.
- Environmental Factors: Temperature, wind resistance, and altitude can also play a role in fuel consumption.
Statistical analysis plays a crucial role in verifying car manufacturers' mileage claims. By conducting random sampling and applying hypothesis testing, consumer testing agencies can provide valuable insights into the fuel efficiency of car models. This information empowers consumers to make informed decisions when purchasing a vehicle. Understanding statistical concepts like average, standard deviation, and p-value is essential for interpreting the results of such analyses. Remember that fuel efficiency can be influenced by various factors, and real-world mileage may vary from the advertised figures.
- Car mileage rating
- Average mileage
- Standard deviation
- Random sample
- Hypothesis testing
- Z-statistic
- P-value
- Significance level
- Fuel efficiency
- Consumer testing agency
1. What does the car manufacturer advertise for the mileage rating of its car model?
The car manufacturer advertises the mileage rating for one of its car models to be, on average, 32 miles per gallon (mpg), with a standard deviation of $\sigma=2 mpg$.
2. How many cars did the consumer testing agency select for the random sample?
The consumer testing agency selected a random sample of 30 cars of this model.
3. What is the standard deviation of the mileage rating?
The standard deviation of the mileage rating is 2 mpg.
4. What is the importance of random sampling in this context?
Random sampling is crucial to ensure that the results are representative of the entire population of cars of this model, minimizing bias in the results.
5. What statistical methods can be used to analyze the mileage ratings?
Hypothesis testing, using the z-statistic, can be used to analyze the mileage ratings.
