Statistical Analysis Of Car Mileage Claims A Consumer Testing Agency Investigation

In the dynamic world of automotive engineering, fuel efficiency stands as a crucial factor influencing consumer decisions and environmental considerations. Car manufacturers invest significant resources in optimizing vehicle performance to achieve competitive mileage ratings. Mileage rating, often expressed in miles per gallon (mpg), serves as a key indicator of a vehicle's fuel economy. Car manufactures commonly publicize these mileage estimates through various marketing channels to entice prospective purchasers. However, the true fuel efficiency experienced by drivers in real-world conditions can vary due to factors such as driving habits, road conditions, and vehicle maintenance.

This analysis delves into a statistical investigation of fuel efficiency claims made by a car manufacturer. We examine a scenario where a 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 $\sigma=2 mpg$. A consumer testing agency, committed to providing unbiased assessments, undertakes a random sampling of 30 cars of this model to independently verify the manufacturer's claim. This process allows us to explore the principles of statistical inference and hypothesis testing, essential tools for evaluating claims and drawing conclusions about populations based on sample data. By employing statistical methods, we can determine whether the sample data provides sufficient evidence to support or refute the manufacturer's advertised mileage rating. This involves calculating sample statistics, constructing confidence intervals, and conducting hypothesis tests to assess the validity of the manufacturer's claim. The analysis will not only shed light on the specific case of the car model's fuel efficiency but also illustrate the broader applications of statistical analysis in evaluating claims and making informed decisions across diverse fields.

To begin our statistical journey, it's crucial to define the population and the sample within our context. In this scenario, the population encompasses all cars of the specific model manufactured by the car company. This represents the entire group about which we seek to draw conclusions. The population mean mileage, denoted by μ, is the actual average mileage for all cars of this model. The manufacturer advertises this population mean to be 32 mpg. The population standard deviation, denoted by σ, measures the spread or variability of mileage ratings within the entire population of cars. The manufacturer specifies this population standard deviation as 2 mpg. The sample, on the other hand, is a subset of the population. In this case, the consumer testing agency selects a random sample of 30 cars of the model. This sample serves as a representative snapshot of the population. We use the sample data to estimate population parameters and test hypotheses about the population. Key sample statistics include the sample mean, denoted by x̄, which is the average mileage of the 30 cars in the sample, and the sample standard deviation, denoted by s, which measures the variability of mileage ratings within the sample. The relationship between the sample and the population is fundamental to statistical inference. We use sample statistics to make inferences about population parameters. For example, we use the sample mean x̄ to estimate the population mean μ. However, it's important to acknowledge that sample statistics are subject to sampling variability. Different random samples from the same population will likely yield different sample statistics. This variability is quantified by the standard error, which measures the standard deviation of the sampling distribution of a statistic. Understanding the population and the sample is paramount in statistical analysis. It allows us to frame our research questions, select appropriate statistical methods, and interpret our findings in the context of the population. In the subsequent sections, we will delve deeper into analyzing the sample data to draw conclusions about the population mean mileage of the car model.

The sample mean is a crucial statistic in our analysis, serving as an estimate of the population mean mileage. To calculate the sample mean (x̄), we would sum the mileage ratings of the 30 cars in the sample and divide by the sample size, which is 30. This calculation provides a single value that represents the average mileage observed in the sample. While the sample mean is a valuable estimate of the population mean, it's important to recognize that it is subject to sampling variability. This means that if we were to take another random sample of 30 cars, we would likely obtain a slightly different sample mean. To quantify this variability, we calculate the standard error of the mean. The standard error (SE) measures the standard deviation of the sampling distribution of the sample mean. It essentially tells us how much we expect the sample mean to vary from sample to sample. The formula for the standard error of the mean is: SE = σ / √n, where σ is the population standard deviation and n is the sample size. In our case, the population standard deviation is given as 2 mpg, and the sample size is 30. Plugging these values into the formula, we get: SE = 2 / √30 ≈ 0.365 mpg. This standard error indicates that the sample mean is expected to vary by approximately 0.365 mpg from the true population mean. A smaller standard error suggests that the sample mean is a more precise estimate of the population mean. Conversely, a larger standard error indicates greater variability and less precision. The standard error is a critical component in constructing confidence intervals and conducting hypothesis tests. It allows us to assess the uncertainty associated with our sample estimates and draw statistically sound conclusions about the population. In the upcoming sections, we will utilize the sample mean and standard error to construct a confidence interval for the population mean mileage and conduct a hypothesis test to evaluate the manufacturer's claim.

A confidence interval is a fundamental statistical tool that provides a range of plausible values for an unknown population parameter. In our context, we are interested in estimating the population mean mileage (μ) for the car model. A confidence interval is constructed around a sample statistic, such as the sample mean (x̄), and is defined by a confidence level. The confidence level represents the probability that the interval will contain the true population parameter. For instance, a 95% confidence interval suggests that if we were to repeatedly sample from the population and construct confidence intervals, 95% of those intervals would capture the true population mean. To construct a confidence interval for the population mean, we utilize the sample mean, the standard error, and a critical value from a probability distribution. The choice of distribution depends on factors such as the sample size and whether the population standard deviation is known. In this scenario, we have a relatively large sample size (n = 30) and the population standard deviation (σ = 2 mpg) is known. Therefore, we can use the z-distribution to determine the critical value. For a 95% confidence interval, the critical value (z*) is 1.96. This value corresponds to the z-score that leaves 2.5% of the area in each tail of the standard normal distribution. The formula for the confidence interval is: Confidence Interval = x̄ ± z* * SE, where x̄ is the sample mean, z* is the critical value, and SE is the standard error. Let's assume, for the sake of illustration, that the sample mean mileage (x̄) is calculated to be 31.5 mpg. Using the standard error we calculated earlier (SE ≈ 0.365 mpg) and the critical value (z* = 1.96), we can construct the 95% confidence interval: Confidence Interval = 31.5 ± 1.96 * 0.365 ≈ 31.5 ± 0.715. This yields a confidence interval of approximately (30.785 mpg, 32.215 mpg). This interval suggests that we are 95% confident that the true population mean mileage for the car model falls between 30.785 mpg and 32.215 mpg. The width of the confidence interval reflects the precision of our estimate. A narrower interval indicates a more precise estimate, while a wider interval suggests greater uncertainty. The confidence interval provides valuable information for evaluating the manufacturer's claim of an average mileage rating of 32 mpg. If the advertised mileage falls within the confidence interval, it suggests that the sample data is consistent with the manufacturer's claim. Conversely, if the advertised mileage falls outside the interval, it may raise concerns about the validity of the claim. In the next section, we will delve into hypothesis testing, a formal statistical procedure for evaluating claims about population parameters.

Hypothesis testing is a cornerstone of statistical inference, providing a structured framework for evaluating claims or hypotheses about population parameters. In our context, we aim to assess the car manufacturer's advertised mileage rating of 32 mpg for the car model. We can formulate our hypothesis test as follows: Null Hypothesis (H0): The population mean mileage (μ) is equal to 32 mpg. This represents the manufacturer's claim and serves as the default assumption. Alternative Hypothesis (H1): The population mean mileage (μ) is not equal to 32 mpg. This contradicts the manufacturer's claim and suggests that the true average mileage is different from 32 mpg. The alternative hypothesis is a two-tailed hypothesis, meaning that we are considering deviations from the null hypothesis in both directions (above and below 32 mpg). To conduct the hypothesis test, we calculate a test statistic that measures the discrepancy between the sample data and the null hypothesis. In this case, since we know the population standard deviation and have a relatively large sample size, we can use the z-test statistic. The formula for the z-test statistic is: z = (x̄ - μ0) / (σ / √n), where x̄ is the sample mean, μ0 is the hypothesized population mean (32 mpg), σ is the population standard deviation (2 mpg), and n is the sample size (30). Using our assumed sample mean of 31.5 mpg, we can calculate the z-test statistic: z = (31.5 - 32) / (2 / √30) ≈ -1.369. The z-test statistic represents the number of standard errors that the sample mean is away from the hypothesized population mean. A larger absolute value of the z-test statistic indicates a greater discrepancy between the sample data and the null hypothesis. To evaluate the z-test statistic, we compare it to a critical value or calculate a p-value. The critical value is determined by the significance level (α), which represents the probability of rejecting the null hypothesis when it is actually true. A common significance level is 0.05, corresponding to a 5% chance of making a Type I error (incorrectly rejecting the null hypothesis). For a two-tailed test with a significance level of 0.05, the critical values are ±1.96. If the absolute value of the z-test statistic exceeds the critical value, we reject the null hypothesis. Alternatively, we can calculate the p-value, which is the probability of observing a test statistic as extreme as or more extreme than the one calculated, assuming the null hypothesis is true. A smaller p-value provides stronger evidence against the null hypothesis. In our case, the p-value for a z-test statistic of -1.369 is approximately 0.171. This means there is a 17.1% chance of observing a sample mean as far away from 32 mpg as 31.5 mpg if the true population mean is indeed 32 mpg. If the p-value is less than the significance level (α), we reject the null hypothesis. If the p-value is greater than the significance level, we fail to reject the null hypothesis. In our example, the p-value (0.171) is greater than the significance level (0.05), so we fail to reject the null hypothesis. This suggests that the sample data does not provide sufficient evidence to refute the manufacturer's claim of an average mileage rating of 32 mpg. However, it's important to note that failing to reject the null hypothesis does not necessarily prove that the null hypothesis is true. It simply means that we do not have enough evidence to reject it based on the available data. In the conclusion section, we will summarize our findings and discuss the implications of our analysis.

In this statistical analysis, we investigated the fuel efficiency claim made by a car manufacturer for one of its car models. The manufacturer advertised an average mileage rating of 32 miles per gallon (mpg) with a standard deviation of 2 mpg. To assess this claim, a consumer testing agency randomly sampled 30 cars of this model. Our analysis involved constructing a confidence interval and conducting a hypothesis test to evaluate the validity of the manufacturer's claim. First, we calculated the sample mean and standard error to understand the central tendency and variability within the sample data. Assuming a sample mean of 31.5 mpg, we computed a standard error of approximately 0.365 mpg. This standard error quantifies the expected variability of the sample mean from the true population mean. Next, we constructed a 95% confidence interval for the population mean mileage. The confidence interval, calculated as (30.785 mpg, 32.215 mpg), provides a range of plausible values for the true average mileage of the car model. Since the manufacturer's advertised mileage of 32 mpg falls within this interval, the sample data appears consistent with the manufacturer's claim. Furthermore, we conducted a hypothesis test to formally evaluate the manufacturer's claim. We formulated a null hypothesis that the population mean mileage is equal to 32 mpg and an alternative hypothesis that it is not equal to 32 mpg. Using a z-test, we calculated a test statistic of -1.369 and a corresponding p-value of 0.171. Since the p-value is greater than the significance level of 0.05, we failed to reject the null hypothesis. This indicates that the sample data does not provide sufficient evidence to refute the manufacturer's advertised mileage rating. In summary, based on our analysis of the sample data, we do not have enough evidence to reject the manufacturer's claim of an average mileage rating of 32 mpg for the car model. The confidence interval suggests that the true population mean mileage is likely to be within a range that includes the advertised value, and the hypothesis test did not yield a statistically significant result to contradict the manufacturer's claim. However, it is crucial to acknowledge the limitations of our analysis. Our conclusions are based on a single sample of 30 cars. A larger sample size or multiple samples could provide more robust evidence. Additionally, our analysis assumes that the sample is representative of the population and that the population standard deviation is known. In real-world scenarios, these assumptions may not always hold true. Therefore, while our analysis provides valuable insights, it is essential to interpret the results cautiously and consider the potential for sampling variability and other factors that could influence the findings. Further research and data collection may be warranted to gain a more comprehensive understanding of the car model's fuel efficiency.